If he's willing, go for it. Don't expect him to win any hunter classes, but just for fun - why not?
I ride at a hunter/jumper barn and we have what we think is a TWH mixed with QH (our best guess) - he does trot and canter, but it's kind of a funky trot and canter, like it's a little bit gaited and a little normal. Or perhaps he switches between gaiting and not gaiting. Idk, but it feels pretty weird. Comfy, but weird. He goes in lessons and can jump up to 2'6", he just never gets taken to shows because the judges would be like 'wtf is that?' :) He's actually really good at making tight jumper turns, and he's super safe for the kids.