The problem will be that with it being a racer, you'll have poor driveability anyway. Might not get anything out of it under about 3k revs. So with an auto you'll need a high stall converter, and the nice OD you want won't get you anywhere because it won't engage the OD-clutch under about 80 mph or something. Running low revs with a high stall TC makes serious heat and doesn't move a vehicle in a specially nice way because of the slip you get across the TC. A well built 700-R4 would get you there (plan on about 2k for the trans), but a stick is the way to go in my opinion (but get a good clutch).
If you do stay automatic, I would go for the 700-R4 because it has a lower first gear than the 200-4R, so it will get you off the line quicker and it weighs a good deal less than the 4L80, and weight is an issue in every racer. The 4L60 is basically the same as the 700-R4, but it's controlled over the ECU of the engine, which you probably won't be running, I guess.
That's as always just me thinking loud...
Greetings from across the lake,
Max