Chill by Alpha Omega works well or if you want fully natural go to health food store and buy a bag of raspberry leaves. Feed a hand full daily or just a few days daily before show
Raspberry leaves are packed with magnesium which has calming properties. Try to choose as natural of a product as possible
Also go to show early. Walk them around and get them more comfortable with the situation before your class.
I think if it makes the horses experience more comfortable then sure but maybe see if the first day of the show you can just lead him around and let him see everything so when you get on he feels more relaxed? I know the feeling with golf carts! One almost ran into my horse and I once at a show because they were going too fast around a corner, he got a fright but was fine
My problem is that when we go to the state fair and big shows at the same place, there are trains right next to us, planes overhead, golf carts zooming passed us and people who are not familiar around horses. The bond and trust built between my horse and I (for 5 years now) is stronger than ever. I have trained this horse since she has 2. This is my dilemma, we do not have any problem with training at home, it is how we should deal with the crazy things that are surrounding us at these places...
In my opinion, a horse that suffers anxiety at shows has not built enough trust in the rider at home. This is a partnership between horse and human with the human being the alpha. It all starts at home. If the horse and rider bond is strong, then the anxiety will pass. Remember , the horse feeds off the humans emotions.