Firstly, just about any battery with enough range to go 150 miles will have high enough output. Back of napkin calculation:
Single seat car with everything in it weighing about 1000 lbs (this will be very difficult) would get 100wh/mile at reasonable speeds. This means to get 150 miles range you would need a 15kwh pack. Since you are talking about higher speeds you wont get 100wh/mile. Lets double that so you would need a 30kwh pack. Lets assume 300 volts which means 100ah cells. If you used 100AH calb cells in a 94S1P arrangement you are talking about a pack that could realistically output 271kw (363hp) for perhaps 10 seconds at a time dropping back to an average of 81.2kw (109hp) or less the rest of the time. Such a pack would weigh more than 705 lbs (320kg) So for the driver and the weight of everything else you have only a 295 lb budget. Doesn't seem remotely practical. If you use Tesla's style batteries in a pack of 30kwh it would consist of 2133 cells (3.7 volts and 3.8 ah) have a minimum weight of 211 lbs (96 kg) Such a pack would be arranged as something like 79S27P and be able to do a peak output of around 78.9kw (106hp) with a sustained rating of about half that. The Tesla style pack would need to be larger to do the power levels you want. But at 211 lbs it would give you a weight budget for driver and the rest of the stuff of 789 lbs.
There are higher output batteries available. Cost is high. RC hobby LiPo packs are an example. A super lightweight minimal car will need a 30kwh pack to make 150 miles. Such a pack will cost around $904 per kwh so a 30kwh pack would cost $27114. It would however be able to put out all the power you need as you can dump these packs in less than a minute meaning they can do an average output of around 1.8 mw (2412 hp) and a peak of 3.6 mw (4826 hp). At least if you believe the advertising ratings. Oh and a 30kwh pack would weigh more than 275 kg (605 lbs).
In all cases I am quoting just the weight of the cells. These are unrealistically low numbers as it does not include the inter cell connections or mounting hardware or the cooling system that will need to be used.
I believe you have unrealistic expectations for your project. You can do the 150 mile range or the high speed but not both at the same time. But it doesn't hurt to shoot high and then back off to something that can be done. You might want to research the Formula E race cars. They probably have the performance level you are wanting and a pretty big budget so you can see what is being done for real.
