CAE in Montreal has an A310 used regularly by Air Transat and others. You might want to check with them to see if it's FAA certified. Give them a call @ 514-341-6780
As far as I know, CAE do not provide A310 Type Ratings but they may be able to find a qualified program in their system that a contract instructor can deliver.
Lots more questions I'm afraid but you can easily direct them to CAE.
P.S. Type Ratings issued by the FAA are A300 (for the B4) and A310 (for the 600)