Two of the TWH's I've owned would occasionally trot under saddle. I've maintained that a TWH or TWH cross will take out their anger under saddle by performing a trot or a "broken washing machine" gait. Certainly preferable to bucking or bolting! When "Tyke" (1970-1998, RIP, TWH/Qh cross) didn't like what you were doing, he would trot. He often trotted when being ridden for lessons and he didn't care for the work, such as jumping cavaletti with a novice.
I suspect bc of his size, that there is a LOT of TWH in my 6yo KMH gelding. Periodically, he trots during schooling, so I just post. He came to me with some fragile confidence and they'll be plenty of time to discourage the trotting in lieu of ambling...or, maybe, "run-walking."
During the amble or the running walk you could post it, but the pace is so quick you would soon wear out. Better to just enjoy a comfortable gait. It doesn't hurt them to have to you sit it, unlike a young or out of shape older horse, or posting bc your horse hasn't warmed up yet.