It's been a while since I used it, but here's what I remember...
A flight school must be approved by the state. Part 141 implies, but doesn't guarantee approval. In theory, a Part 61 school could be approved, but I don't know of any.
Unless things have changed since 1993(the first time I used my benefits), you must apply for benefits for each training course. I had to apply for the benefits for my CFII and MEI seperately.
Start with the 141 schools in your area. If you don't like what you find, keep looking. American Flyers in Addison, TX is where I used my benefits. I haven't seen any of their ads recently but I believe all their locations are approved for GI Bill. Quantico MCAS was approved.
Fly safe!