Thank you everybody for your responses.
I have 3 follow up questions:
Stephem: You mention the routing option of TPA-DFW-FCO which I had not considered. If I want to pick this one as an example, how would you go about finding this routing availability?
Guv1976: Yes, you are correct in that BA does fly TPA-LGW-FCO. Given that I have triple the amount of AA miles compared to BA miles, is there a trick to finding BA flights using AA miles?
General Question: If I use miles from TPA-LGW in business and buy a ticket in cash in economy from LGW-FCO, would I be able to check my luggage through to FCO?