Triumph managed to engineer the old "Knock-Wobble-Bang" (followed by a pool of oil) out of their designs back in 1972, so there shouldn't be anything other than a bit of vibration through the pegs. What you're describing is certainly not a feature!
I agree with muensterDeath, it would be difficult for the stock cat to move as there's a post that inserts into a rubber grommet on one side, two bolts on the other, and four exhaust connections. Everything would have to be seriously loose for it to move around...and your exhausts would be noticeably blowing.
It could be one of the four exhaust clamps that grip the pipes onto the cat though, and it's easy to check if you lie flat on the floor.
Are the removable frame rail bolts torqued tight (also easy to check)?
How about the bolt securing each end can to the frame (it screws in from underneath)?
Also worth checking the frame brace in front of the engine.
Unlike the right side, the left hand exhaust system has a removable plate that bridges the header pipe and the tail pipe where they enter and exit the cat. It's a push fit into the header and a clamp secures it to the tail pipe. The clamp may be loose or may have fallen off completely.
I would have imagined that these are all obvious parts for your dealer to check though.
Hope it helps though, but let us know what it turns out to be!