No one likes to hear this but the underlying cause for this is your horse doesn't trust you enough yet in situations that are stressful for him. So any kind of trust building excercises like ground work, lunging, hand walking him will probably help. And to support your effort you can always try a combination of magnesium and vitamin b supplements.
sounds like the problem is separation anxiety. This will probably require lots of time and patience. If you have the opportunity you could start by getting him used to it in small steps. Put him on the trailer and when he relaxes (or before he get's agitated) take him back to his stall/pen. If that is no big deal anymore drive a short distance and then put him back, over time you can increase the duration. Maybe you have another barn close by where you can trailer him to, ride there and drive back home.
He's jumping and kicking a lot in the trailer. Also if his neighbors horses goes out for running pen or riding he starts to be crazy in the stall. I'm thinking it will be uncomfortable for him in the temporary stable at the shows. Is it clear what I'm worried about?
How do you know he's not keen on travelling? Is he jumping around in the trailer, does he sweat a lot, is it just the trailering or his behavior at the venue as well? There's different things you can do depending on what his issue is.
Is your horse really afraid or travelling or is it just that he doesn't really enjoy that but can bear to travel?
Because if he is really afraid, you will need a lot of patience to make him understand that you doensn't want to hurt him.
You should start with very short trip just to get him used and when he is more relaxed you can take him on longer trips.
Drive very gently and frequently offer a brake when you give him a treat and go see him.
You can also try to give him homeopathic products to help him relax.
Good luck with your horse, hope this helps :)