You can get a turbo that will spool before 3700 RPM. But here's the thing. But be aware that with the automatic transmission is going to kill your 60 foot times. ShadowDawn has an automatic LS with a RevHard Stage 2 turbo kit running 8 psi. Here's his latest dyno after he tuned the car with an S-AFC:



Good numbers power-wise, of course. Before the car was tuned he ran a 14.8 @ 95.7 mph (this while it was putting down about 175 whp). GSR's with bolt ons can run in the high 14's all day long, although they trap at lower speeds since a typical GSR with bolt ons will put down 155 whp. Granted, traction is the main problem with the automatic and from a rolling start it would probably kill a GSR with bolt ons.

If you don't mind putting a ton of time and money into your automatic car, and don't care about resale value down the line then go ahead and get a stick shift swapped in. Realistically though, it would be much simpler to put your automatic car back to stock and buy something with a stick. Assuming it's an Integra your parts will all swap over or otherwise you can sell them.