I think that it depends of two things.
First of all, it depends of the morphology of your TWH : in general, they have quite a long body, when in barrel race horses generally have quite compact morphology. But maybe your TWH has an intermediary morphology. Moreover, I don't think that TWH are fast enough horses if you compare them to QHs for instance.
The second thing to think about is the level that you want to meet barrel racing with your TWH : if you want to barrel race for fun and to change your routine, I think that you can do it with any horse, as long as you do it correctly, doing the appropriate exercices to prepare your horse, and not just running around barrels. You can't teach alone a horse to barrel race if you never did barrel race but if you have an suitable trainer, you could perfectly have a great time barrel racing with your TWH for fun. I have seen some videos of TWH barrel racing, and they can do quite well. They are just not able to compete with good QHs in serious competitions. That is why to me, if you want to compete and reach a good level, maybe you should try with a more suitable horse.
I hope that my answer helped ! Have a nice day and don't forget to tell us if you finally succeeded in barrel racing with your TWH :-)