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My original thought was to use a zero bike motor and mate it to the stock MR2 transmission. Would make for a generally easy swap, but it doesn't quite get me to my power goals. Also, afaik electric motors tend to destroy manual transmissions when shifted (more on this later).

Then I considered using two of these motors and just mating them to the axles since the bikes are direct drive and good to 124 mph as is (I realize that's partially because of the reduction from front to rear sprocket).
Don't underestimate the importance of that gear reduction, which is substantial. If you go with a single-ratio drive, it still needs to be a significant reduction. Production EVs typically run a 6:1 to 10:1 ratio.

Also, even two the of the Zero motors would be inadequate for a car like this.

Has anyone developed a shift controller that rpm matches the motor to the ground speed for a given gear? Wouldn't that remove the shock load and allow for clutch less shifting? I would imagine that would be fairly doable via a microcontroller and switched gates for the shift knob. Depending on which shift gate is triggered it would calculate and adjust the motor rpm until the gear is engaged (heck you could use the clutch start safety switch to engage the rpm matching mode). Would this still cause a bunch of shock to the transmission? I would imagine it could be fine tuned until it doesn't but then why has no one done it?
This would work well, and would eliminate the need for both the clutch and synchronizers. It's beyond what DIY builders generally attempt, and even beyond what many understand.

This would be easiest to implement with a sequential shifter (such as in a motorcycle or automotive race transmission) since the direction of the shift (up or down) and the target gear (always the next one, never skipping ratios) would always be immediately known.

Shifting is pretty pointless in an EV. If the motor can deliver all the torque from zero RPM, and stilll adequate torque at high RPM - you've no need to shift while driving at all. Might be advantageous to put it in second instead of third if you're doing an off-the-line drag race, but "fixed" reductions can work just fine with most EVs. My project is going to be left in third gear almost all the time when it's built (I'm doing motor reverse too, so I won't even need to shift when it reverses...)
There seems to be a tendency to think of "all the torque" (meaning the motor's peak torque output) as infinite torque - it's not. With a high-voltage system it is normal to be able to produce the maximum rated power over a broad speed range, so multiple gear ratios are not required. With low-voltage systems based on brushed DC motors of moderate size, it is normal to need to shift to keep the motor in its relatively narrow peak power band over the range of road speeds, although only a couple of ratios is usually enough.
 

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The stock 4-AGE which came in the MR2 made 115-130 HP (depending on year) and 110 Ft•Lbs. So a single zero motor is about on par with the stock engine.
Those numbers look reasonable for a 4A-GE, but unless there is some higher-powered version of the Zero than I found, the zero motor is not even close to that. The original (DS) and higher-power (SR) models of the Zero appear to have 44 and 67 horsepower... way short of any 4A engine. Torque appears to be 106 lb-ft for the SR, which looks close but is presumably attained at only low motor speed. Two Zero motors might be suitable.

What are my other options for motors? Should I be trying to canabalise a production vehicle like the leaf or bolt? I'd rather go with an AC motor, but I'd like some input on which motor I should be looking at, and why.
If you are willing and able to deal with the issues with controlling the motor from a production EV, that's definitely the most effective reasonably-priced source of a suitable motor for a car. It will also come with a single-ratio transaxle, giving the option of using the complete drive unit (motor with transaxle); that would mean dealing with mounts and axle shafts, but not needing to adapt a motor to transaxle or the motor shaft to a transaxle input shaft.

The Leaf is the most common affordable EV, and it has a good well-proven motor. Some conversions have worked out ways to successfully control the Leaf motor outside of Leaf.

The GM EVs (Chevrolet Spark EV and Chevrolet Bolt) have the advantage of being configured with the motor and transmission on the axle line, making the whole package more compact and leaving more space between the firewall (behind the driver) and motor for everything you need to fit in (such as a battery pack). Of the GMs, the Bolt has a more powerful and smaller motor; it will be more common, but is newer so the salvage supply of Spark EV motors may be greater for now. Here's a successful project using the Bolt drive unit (and battery - but you won't be able to use a Bolt battery in an MR2):
Westfalia T3 with Chevy Bolt drivetrain
 

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Well, I was interested in using two motors for exactly that reason. I don't like open differential vehicles. Two motors allows me to emulate a performance differential, but then I only have a set gear reduction which may not necessarily be a bad thing.
It's certainly an effective way to go, but not trivial to implement.

Does the Tesla transaxle have torque vectoring, or some form of limited slip differential?
There are a few different Tesla transaxles (Model S and X are different from Model 3, front and rear have detail differences, and small-motor and large-motor Model S/X rear units are different), but none of them have any form of limited-slip differential and there is no way for any of them to apply different torque to left and right wheels... other than individual wheel brake application as in every other current production car.

It is possible to fit a different differential in a Model S transaxle, and there are even commercially offered Quaife units to fit. I haven't heard of anyone going to the trouble of doing this for any other EV transaxle, just because the Teslas have been around for quite a while (so they're commonly used) and are especially popular in high-powered conversions, but almost any differential can be swapped out for a Quaife or similar unit.

As for the batteries, the MR2 fuel tank was in what would normally be the transmission tunnel. So a few could probably fit in there. Maybe not 80 miles worth, but at least a couple. The engine bay is rather large compared to an ev transaxle so probably a couple could go in there. It also has a frunk and a trunk. I'd like to keep enough storage capacity to pick up groceries, I'd also like to keep the vehicle as close to a 50/50 balance as possible (I have plate scales to help achieve that).
The tunnel space is great - well-located for mass distribution - but modules are rigid things and few if any are sized to fit in that tunnel. Of course the Chevrolet Volt (and Volvo hybrid) modules are designed to fit in a tunnel, but they go in cars with a tunnel designed to accommodate them. Do you have the dimensions (particularly width and height) of the tunnel space?

You might be surprised how large a typical EV drive unit is. Open the hood of a Leaf: the drive unit is as tall and wide (front to back) as an engine with transaxle; it just isn't as long (across the car). The inverter and charger are a substantial part of the height, but you need to those things somewhere. This is why I mentioned the Chevrolet bits, which sit further back due to the concentric design.

The rear trunk is bad for mass distribution (you don't want it at the ends of the car), but the front trunk seems like a good place for a small pack.

An MR2 isn't a 50:50 balanced car; it's rare for a rear-mid-engined vehicle to not be rear-biased, and as a rear-wheel-drive car designed to be rear-heavy, that's okay. It's slightly rear-heavy when empty (about 45:55, depending on roof style and engine, for the W10 generation), driver/passenger load is also rear-biased, and of course cargo load capacity is much greater in the rear than the front. Some change is manageable, but even without driving one I'm sure that I wouldn't want a front-heavy MR2, and even 50:50 wouldn't be an improvement over the stock distribution.
 

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A slightly out of the box idea would be to consider mounting the motor in the fuel tank tunnel, with a driveshaft running through to a standard RWD differential (from an IRS rear end, not live axle).
... it would also leave a lot of the engine bay free for batteries. That would keep the bulk of the new weight in about the same location as the original layout, and you could move the motor fore or aft as needed to help achieve the target weight distribution. Obviously this option depends on the motor being physically able to fit into the tunnel, and I haven’t seen many of the popular EV motors in the flesh to have an idea of the physical size of them.
I'm pretty sure that someone has built that configuration. One limitation is that the tunnel is not huge - it will limit motor diameter; production EV motors are big. And the space between the axle line and firewall will be substantially compromised by the shaft and the nose of the final drive.

Here's an example in EV Album, from a decade ago:
1987 Toyota MR2 AW11 “Black Magic”
This one apparently used a Mazda MX5 (Miata) diff, which is the same Hitachi as used in many Nissan and Subaru models.

I was surprised to find about fifteen MR2 conversions - of all generations but mostly W10 - in the EV Album. Generally they adapt a DC motor to the stock manual transaxle and stack some battery above it, with LiFePO4 cells (since none are recent), but there are some AC motors and even some lead-acid batteries. None of the others use the motor-in-tunnel solution.

This would allow you to get a decent gear reduction (I guess if you search hard enough you may find a suitable diff with a ratio of 5:1? I think I remember hearing there were some quite short diff ratios available for the Toyota GT86)...
The GT86 (and Subaru BRZ, and Scion FR-S) use the same final drives as some other rear-drive Toyota models (despite being built by Subaru and based on the Impreza); apparently 4.10:1 and 4.30:1 have been stock but 4.56:1, 4.67:1, and 4.88:1 are all available. Apparently the big and strong Toyota diff is not the one in the GT96, but the 7.5" F-series, and there are some more extreme ratios of that, but those ratios will be in obscure models and likely difficult to find... but still a possibility.

I don't know which diffs would be easiest to adapt to MR2 axle shafts.
 

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I like the idea, it gives me an idea that I think is even more useful. I'd try and mount the motor up front, then just use the trans tunnel for the driveshaft.
That means not being able to use any of the space in the front or in the generously-sized tunnel for battery, and still having a final drive unit (differential) in the original engine space, keeping the battery pack from being able to use all of the space to the floor level. This seems like a strange way to use a vehicle designed for a mid-mounted engine.

Using the engine bay for batteries does seem to make sense since likely, unless I chose to have very short range, there will be more weight in batteries than motor.
I agree with this part.
 
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