Sounds like you need a polo horse!
I rode VERY BRIEFLY on a polo team (did not like how they treated their horses and left).
Have the horse in a round pen, and just stand there waving a bat around in the air. BE SURE NOT TO HIT YOUR HORSE.
Wiffle bats will not quite be long enough. You will need to be able to ride your horse single handed. If this game will work how I assume it will (similar to polo) the horses also need to be comfortable with ramming into each other, and ridding side by side and pressing each other into the wall.
Get the horse used to the bat, then slowly rub it all over him. Once he ignores that, lightly tap him with it on the back, stomach, and up by his neck. So he hears the noise and realizes it won't actually hurt him. Then get on him (assuming he is a quite horse) and rub the bat around his neck and flank. Swing it slowly up by his face so he can get used to it being there, and popping into his line of vision sometimes. Then begin walking around the round pen, and swinging it slowly around up in the air and around his face. Then reach down while he is still walking, and let it bump into his legs as he walks. Swing it around his legs. Wiffle bats make noise when they swing, so get him used to that noise happening all around him.
As for the beach ball, just put him in the round pen, and casually toss it around, roll it towards him, and see how he reacts. Basic bombproofing should help, have him walk on tarps, put tarps on him, etc. Roll the beach ball around. If it's like a $1 beach ball, get on him, and try to have him walk into it. You need a horse that will not shy from the ball. If he steps on it and pops it, oh well (another problem you will have with beach balls). But he should be aware that if it pops, which may happen, it will be loud.
Sounds like you guys are putting together a ghetto version of polo. Just do polo.