If your horse is only part TWH, it would be do-able. I worked with a 7 year old random crossbreed gelding who was a part gaited thing, Missouri Fox Trotter was our best guess. He gaited when he was born, then grew out of it and did walk/trot/canter at liberty. He was green broke and I just helped put more rides on him. His owner just wanted him to be safe for her nieces and nephews to ride and be able to do intro level at schooling shows, so she wanted him to trot. All I did was ignore the gait and keep posting to help him keep the tempo. I would often click or tap the whip in rhythm and praised him hugely when he would trot. He really just got his legs mixed up a lot of the time and didn't know where to put them. After a month of riding him, he would trot and canter normally 95% of the time.
If your horse is full TWH, I would not try to change him. He was bred to gait and changing that goes against what he was built/bred for. Find a barrel horse if you want to run barrels.
"He doth nothing but talk of his horses."