There are so many things to address here I didn't even try quoting, because the quotes would dominate my whole post. I'll try to just address subject matters, instead of exact wording or posts.
Autocross
I am building an autocroser as well, and also plan to sprint down the drag strip. What kind of autocross are you running in that lasts 5 minutes!
That seems like an awfully l-o-n-g time for an autocross race, and more like a short road course. The autocross races I watched at the local Goodguys car show here in Columbus were completed in less than 30 seconds by the fastest cars, and still under a minute by the slower ones.
The biggest point with this subject, in your case, is weight. Even a stripped down unibody car is going to be too heavy to effectively compete in an autocross race without a transmission - or - rear end gearing that is too steep to do anything more than city driving. Those gears may also come back to bite you in the autocross because it won't want to stretch out on the final straight. You will punish your motor and batteries running direct drive, because off the apex of every turn you're going to be asking them for max current to make up for not having enough gearing.
My project, by comparison, is being scratch-built and purposely designed to run direct drive. I am shooting for a final weight of 1200lbs with A123s. I am also using an 11-inch motor, which is capable of a ridiculous amount of torque for a 1200lb vehicle. The goal is that I won't be stressing the motor and batteries so much off the apex because it will actually have more torque than I can succesfully apply. The vehicle is also a dedicated race and show car, with no real-world expectations other than an occasional trip up the street for fun. That allows me to (hopefully) afford A123s. It looks like my pack will be around $15K, and under 500lbs.
I don't mean to go on about my project. I am trying to give you an idea of how the project was planned. Unless we are missing critical details, your plans seems to be a mix of components and ideas that will likely result in disappointment. You have a production street car that you plan to convert to an EV racer/driver, with a mild performance-oriented powertrain combination. It sounds like you will ultimately break a lot of parts, and eventually kill the motor and/or battery pack. Shoving enough current through to repeatedly accelerate the steel-bodied car will stress the batteries and motor, and the high torque loads delivered electric motor style (instantaneously - like a hit of nitrous oxide) will stress the driveshaft and rear end.
Transmission
As noted, running direct drive with a relatively heavy vehicle is probably asking too much from your motor and batteries - in an autocross race environment. If you were just drag racing you could manage it. The converted Powerglide (do a search here) might be a nice option if you don't want to shift, and want a more futuristic "feel". They can be built to withstand over 1200lb-ft of torque and live to tell about it. There are also a number of racing manuals that can handle high-torque racing applications. When I was considering using a transmission, one thing I found appealing is that you can use dog-engagement race transmissions that can be shifted without a clutch. Because an EV can simply select the appropriate gear from a standing start and accelerate without use of the clutch it opens the door for using these transmissions. Whatever route you plan to go, if you choose to use a transmission, be prepared to pay the piper. You need a really stout transmission race transmission because the street models aren't designed to handle the repeated shock of instantaneous torque hits. The phrase "nitrous-ready" is a good thing for an EV racer's parts.
Controller
I'm not pitching for anyone - yet - but have a couple favorites. I have been enamored with Zillas since I first learned about EV drag racing, and then along came a spider named Soliton. I like the Zilla's track record and relentless ability to slam as much current as your batteries can muster up on the motor. I like the Soliton's hi-tech build and ability to deliver whopping doses of current until someone screams "Uncle!" I am split between which one makes the most sense in an autocross race. Until there's a 2K Soliton, the Zilla will win likely continue to dominate at the drag strip, but autocross is a different animal. It's unproven whether one could even get 2K amps into the motor and on the ground, and you're definitely not going to use any substantial amount of current continuously. You're on and off the pedal, from apex to entry point. I watched some 500-700hp muscle cars (with road-race suspension underneath) go through the cones at the aforementioned Goodguys event. They struggled to successfully apply the ramp up of torque their ICE powertrains delivered - even with really wide, sticky, rear tires. The ones who got a little overzealous with that application ended up doing nice little 360s. One guy just enjoyed the moment, kept his foot in it for a couple turns, then emerged from a cloud of tire smoke. This would be even more challenging with an e-motor that's capable of full torque whenever you ask for it.
I look forward to that challenge but, as I pointed out, spent a lot of time designing for the intended mission.