Either you have thoroughly misunderstood the OP or I have.

I understand he is using the BTS website to work out which particular aircraft he has flown.

The BTS website lists registrations for most airlines, but for AA it lists a strange combination of AA's fleet number for that aircraft, and attempts to format it into an FAA registration format.

So from the OP's example, the fleet number would be R01.

He wishes to find a resource that will tell him what the FAA registration of fleet number R01 is (as an example. I don't think R01 is a valid fleet number, hence my example of a 777) for which I have provided him with some guidance.

The FAA website has no record of Fleet Numbers, as these are an entirely internal feature of an airline, and don't have any need to be recorded with the FAA.