Is this pee problem recent or did he always have troubles keeping his pee? If the problem started at one point, do you remember more or less when did it start?
It’s just a hypothesis, but if he’s being treated with antibiotics for a long period of time, it can be he’s having an urinary tract infection. Or it can be the syphilis has weakened his uro-genital muscles and if that were the case, I think there’s little to be done.
Being the slave of a very delicate bun health-wise, I’ve learned to stay put and be more relaxed about it. I know his medicines, I know the doses and I know what to do when he has certain symptoms, so that if anything happens I always can wait a day or two before harrassing my vet. If you can handle bunny anxiety, you’ll be fine with baby anxiety! (at least babies learn to talk!)
Since you’re busy and you must be very stressed already since your baby is coming, have a phone call with the vet to be reassured about his health, buy some absorbant sheets for puppies, tape them down to the places the bunny relaxes (like in the cardboard house) and cover them in fleece (so that the buns won’t chew them). In case of an accident the buns won’t get dirty and you won’t have rabbit pee flooding your house. I wish you all the luck!