Hmm.... Here's what I would do if this was my situation...
Assuming that the dog is a secure dog that is not suffering anxiety from being alone or separated from its people, I would first remove all bedding from the crate. I'd probably put an old, clean towel in the crate until I feel secure that the crate-soiling is a thing of the past.
I'd also move the crate to my bedroom next to the bed. I'd make sure that the water was picked up several hours before bedtime, and I'd be sure to take the dog out before bedding down. I'd stand there and watch, making sure the dog pees. As I put the dog into the crate, I'd introduce a command that should be associated with settling down for the night. With my girls, we use "go to bed."
During the night, I would listen more for stirring and scratching (with just a towel, the dog's claws should scratch the bottom of the crate, making a lot of noise) than for whining or barking. When I hear the dog moving about, I'd take it out. I would not speak to the dog during the process UNTIL it eliminates outside. Then, I would gently praise/cuddle the dog for going outside. We'd then return to the crate.
Once we lasted a week or so without any crate accidents at night, I would try to settle the dog vocally with repeating the "go to bed" command when it starts to make noise. Sometimes reaching out to the dog through the kennel helps settle a dog. If the water's brought up and the bladder is relieved before bedtime, and the dog is otherwise healthy, it should be able to control and hold it until morning. If the dog can not settle, or only settles for a brief period of time, I'd take it out as described above. The dog should, after time, adjust and sleep through the night. (If it doesn't, I would probably go back to my vet and explain the details. Perhaps there's something else going on besides a bladder infection.)
Only after a week or two of all-night sleeping without any outings or accidents would I return any type of special bedding. I would also get new bedding, probably. The new bedding would avoid any remaining scents of old accidents plus it would be a great reward for me (as dog stuff is FUN to buy) for successfully getting through this challenge!