I would first check the rules of where you are showing since everyone can vary. Most of the shows I have been to allow you the option of a bridle with reins OR a plain leather halter with a matching lead for English Showmanship. Western Showmanship is almost always going to be a halter and matching lead.
I have always preferred to use a halter and lead because it makes it easier for my horse and I to switch from English to Western Showmanship. Your horse will be less likely to get confused if you keep your tack consistent. My horse is a showmanship whiz, but when we do showmanship in a bridle, we're not a fluid and on point with our moves.