Be grateful he will go so he is not damaging his kidneys! A horse who NEEDS to go but won’t, turns into a problem fast. This ‘trailer training’ is actually an asset you can use to have him go before a lesson, at a show before competing, so horse is not distracted by his need to go. When horse needs to go he is not giving you his full attention, not performing at his best.
LOTS of horses won’t go as often as they need to at shows, under saddle, so you can have ‘issues’ where horse is inattentive, unresponsive to signals. Get him to go, everything goes back to normal!
DD’s old horse was both stall and trailer trained to go, so she would put inside either place, every two hours at a show or competition. Give him a couple minutes and he would go. He came back out ready to ride, no distraction issues in listening. He didn’t like being ‘venerable’ to other unknown horses in that stretched position. Previous life as a herd stallion made him a different kind of thinker, but not hard to live with.
Sure made him easy to use, emptied him before use for excellent performance!
Maybe the bucket of sawdust would work well, sure worth trying to save trailer cleanup time. Other wise I would just thank the Lord he is not one who WON’T go, to cost you Vet money. Friend had one like that. He gradually got worse, blew a couple 2 day show fees to take him home after NOT going for 20 hours and more!! He finally would only go at home. She could only take him to short day outings nearby, until she gave up hauling him at all. That not-going stuff will ruin you fun!
If this is his only unlikable trick, you have a really easy time of it with him! I would love a horse with this habit, since bedding and mats get removed after each haul anyway to save the wood floor. I want mine to go anytime they think they need to. Better they go than suffer from holding it in too long. Or one not listening as needed, by being distracted during their work with a strong need to pee. Cleaning the trailer after a show with DD’s old horse, was just part of “going to a show” clean up and put things away procedures. We had heavy bedding to prevent splashes, he liked deep bedding for that.