IMO, turning a cutting/reining horse into English may be a challenge. He's used to a long rein, but in english you have constant contact on his mouth. He may very well dislike the whole idea and constantly toss his head. Another English thing is collection. Cutting/Reining horses are used to being all stretched out (for better lack of term..).
If it were me, I wouldn't change him. But if you're just looking to try it, then by all means, try it! Or instead of messing with your horse's mind, why not try a lesson barn and learn English that way (on English lesson horses)?
Sorry if I miss read your post, please correct me if I am wrong about anything.