If you're from a VWP (visa waiver program country), then you won't need a visa at all as your conversion will take less than the 90 days allowed in the country on the VWP.
The TSA clearance has an option for selecting VWP in the drop down box when submitting your training request, just enter your ESTA number in where it asks for visa number.
You'll also need to get your fingerprints taken for the TSA clearance, and it's worth getting your the FAA license verification done prior to arriving in the US for your conversion as it can take 1-2 months.
Also worth getting an FAA medical done prior to arriving in the US.
I did my ATP conversion at Cloud9 helicopters in Florida. Their aircraft are great, the R22 is $300/HR all inclusive. I only needed about 3-4 hours prep in the R22 (coming from 412's). They'll only do what's required and then put you up for the test. The prep time also included getting me ready for CFI and CFII flight tests. I did all the writtens I required prior to arriving at the flight school.
If other schools try to convince you of minimum hours then they're talking out their ass. There's only a 3 hour minimum prep for CPL tests which you're not doing anyway.
Hope this helps.