As far as your pictures show, I see nothing to discredit it being a T65, with just one codicil (which I'll come back to later).
I imagine the T65 as a way to fulfill supply contracts at that most difficult of times (BSA going bust, Triumph struggling desperately) and so in some business sense, a BSA badged Triumph solved an immediate impasse. Cheekily, I'd suggest that to get Triumph motors may have been a relief to some recipients!
This may also have provided Triumph with an opportunity to use some remaining 650 parts, when they were moving to 750. So this machine should be a 650 to be genuine.
650 barrels have 8 fins. The pics may not be good enough to be certain, but I can only count 7 fins. If true, this would indicate a 750 barrel. Worth checking I'd say.
Of course, pics of the engine/frame numbers would be valuable.
Trivially, the tacho is at 90deg and there is no cable connecting it and its drive gearbox.
The TR65 didn't appear until about 10 years later, again a bit of a last gasp, this time for Triumph. It was almost entirely TR7 other than a shorter stroke crank, another fin off the barrel, and the silly cost savings mentioned earlier.
It was a fine machine, only needed EI and a splayed head to make it fly.