Honestly it sounds like it could be E-Cuniculi, This requires a special blood test, but that often comes up as a false negative, so it’s a very unreliable test. A positive is only shown in bunnies in the top tiers of the infection. Kidney issues are common with EC as it tends to affects the kidneys and brain/neurological functions. My bunny Bombur has chronic EC and kidney issues related to it, and peeing himself and his cage was the only sign we really had that something was wrong. We assumed at first it was behavioral because, like you, he didn’t pee outside the box when he had play time. He also had been urinating and drinking excessively, and still does during a really bad flare up.
You can ask the vet to treat with a standard 28 day course Panacur or Oxibendazole if EC is a possibility. The drug won’t do any harm and will only help him if he does have EC. The drug takes a week or two to fully take affect, so you wouldn’t see immediate results. The drugs can also cause low white blood cell levels, so you’d need to have a blood test at 2 and 4 weeks of treatment to make sure everything is in order.