Help with puppy peeing OUTSIDE of his crate

Jasper is 4 months old. We got him at 10 weeks, and he came to us with a UTI. He has had UTI's on and off since then. He also has had an infection under his foreskin, which also resulted in urgency and frequent urination. Housebreaking has been challenging.

Two weeks ago he had his area "flushed" and has been on antibiotics since then for the infection. I still need to take him to potty every 30 minutes when I'm at home (and he goes every time). If I do not take him this often, he will urinate on the floor. He does sleep for about 4 hours at night before needing to go out.

Three days ago, he discovered a new "trick.' We crate him during the day when we work and have a student coming every 3 hours to let him out, play with him, etc. He has started peeing OUTSIDE of the crate, while INSIDE the crate within the 3 hour window. I sure wish I had a security camera to see HOW he is managing this (it's a good thing he's so cute).

I guess my question is, do you think this is related to his UTI's and he still is not able to hold his bladder for 3 hours at a time? Please tell me eventually he is going to get housebroken. I feel like crating him is not doing any good if he just pees while inside it. We are planning on taking him to doggy day care at the end of the summer when our student sitter goes back to school. Perhaps it's time to transition him to that now, where he will be outside most of the time and can eliminate to his heart's content?

Please give me some guidance and hope!