I can think of how to get 2, but not 3 carriers. (2 will actually save you some money)
LAX-YYZ on AA
YYZ-GRU on CP
book and ticket separately to get the benefit of the lower $CDN and you'll save money.
If you want to save even more, fly CP from LAX-YYZ, then ticket AA YYZ-MIA/JFK-GRU-MVD--that is only $2800 US in business class, $3200 in first class!