Below are some i'm familiar with... You can use the search function here to read what others think. But I recommend you call the schools.
US Aviation Academy
Transpac Aviation Academy
Flight Safety International
AeroSim
Aviator College
CAE (which you know)
WestWind Academy
SAA
http://www.saaintl.com
Treasure Coast Flight Training (online search)
These are just a few. Do an online search or use JC search function and you'll find plenty. Theres one in DC Maryland area as well. I can't remember the name. Also one in New Orleans area of Louisiana
http://www.bestaviation.net/school/flight_academy_of_new_orleans/ .
Florida, Arizona, California, and Texas, New York has some or the larger Flight schools that train International and domestic students. Flordia is like the capital of flight training in the U.S. there are many flightschools in that state.
I know in New York City area there is an Academy of Aviation @KFRG they have int'l students program. M-1 visa assistance I think.
Look online or call these places. Its not easy. But research and ask specific questions, and you'll get answers. Again do some online research, both online forums, and read flight school websites. Plenty of information here on JC.com to help you make an educated decision beyond a glamours website with shinny jets. Ignore the marketing color & glamour plus the "DO YOU DREAM TO FLY" in big bold letters. Ignore the JET, cool pilot glasses and "I'm an awesome pilot in uniform" videos.
Here use this think also in your search
http://www.bestaviation.net/flight_school/
Not all, but some do help with M-1 Visa