If there’s a chance that the shavings are irritating his condition, I would switch to softer litter. I used Carefresh – the white stuff, when Spockie had something going on and I had allergies to some litters. It works well but is softer, so he might find it soothing as he finally heals. It’s a bit more expensive, but not that much.
Both my bunnies, if they didn’t like their litter, peed outside the pan. I doubt that’s why he does it, but switching couldn’t hurt.
He sounds normally hormonal to me too. If his condition makes it unsafe to neuter, you may have to live with him as is for a lot longer. But if he is a single bunny with no chance of meeting a girl, he can be fine. Continue to treat the UTI until the vet gives you the green light on the procedure. If you think your vet isn’t up to it, or just want a second opinion, check out the list of Vets in the post pinned to the top of the Q&A forum section. You might need an exotics vet who’s a bit more experienced with rabbits.