would it hurt him to start barrel racing with me?
Well I took a look at your other post
because I needed more information on your horse.
Well, since he isn't even your horse yet
and since you haven't ridden him yet, it's hard to answer your questions.
In general, TWH don't make good barrel horses (or even a cross). Now, that doesn't mean you can't have fun with a TWH on barrels at a local show. But if your goals
are to be competitive, then I would look for a different horse.
Overall, the horse's conformation is not what I look for when I look for a barrel horse. He is weak in the hindquarters, hocks are high off the ground, looks heavy in the front end (and possibly downhill), and judging by your pictures, appears to have a long back.
Can you have fun with him at local gymkhanas? Sure. You can do so with any
horse. Again, just depends on your goals and how competitive you want to be. Or if you just want to have fun.