The OP implies that their PUJ-BDL flight was canceled, and s/he was put on a PUJ-MIA-BDL connection instead. Since AA doesn't fly PUJ-BDL nonstop, sounds to me like maybe this was at one point a thru flight, then later became a true connection after a schedule change?
Nonetheless, if the OP in fact purchased a business class fare and ended up having to fly in coach on one of the legs, AA is very good about refunding the difference for that leg. Actually, I thought this was automatic and didn't even need to be requested, but a simple letter to CS should suffice. Going straight to the CC company seems like a bad suggestion if they haven't at least tried AA first (other than the PUJ ticket counter).
If, however, the OP purchased a YUP fare in this market (if there are any) and ended up getting rebooked in coach on one of the legs, well that could be a different story...