I'm not a vet, but if the rest of his signs are that good this early, I would lean toward thinking the incontinence will improve. My dog's physical therapist has seen a lot of these cases and she says you have to give it at least 3 months to see if bladder function will return, and ***6 months*** to be sure. That being said, there are examples on this forum of dogs who began to regain continence even later, so that is just a guideline.
It seems like if he has a flaccid anus, he would be easy to express, and you would not have to catheterize? Was there a reason why they went with the catheter instead of simple expressing? I'm guessing there must have been a reason. One thing to keep in mind is, as he recovers, the ease or difficulty of expressing him may change, so if they found him difficult to express right after surgery, that may well change. One tip with expressing male dogs is to hold the pressure for a full 10 seconds (it's a long time) before moving your hands and trying another location.
There is an article with a lot of other tips, plus VIDEOS at the end, that may help if you decide to switch to expressing. There is nothing wrong with catheterizing, it's just easier the other way. One of the moderators here catheterized her dog.
There is a way to stimulate a dog empty the bowel at a time and place of your choosing. This helps avoid the constant cleaning up you may be experiencing. It isn't necessarily 100% but it helps a lot. Here is an article that describes different methods to accomplish this. Many people like the ice cube method or the Q-tip method. I have mostly used the pinching method.
The other thing is to ensure he has quality stools if he is apt to doodle in the house during recovery. You want stools that are firm, low odor, not sticky. You may need to adjust his diet if you are not satisfied with stool quality. I've had really good luck with Science Diet w/d dry (not canned) with my incontinent dogs. I haven't had to use spot remover in years.
If he is able to walk and is dropping surprises around the house, I would really recommend getting a baby gate and keeping him in a part of the house that is easy to clean up. This situation is likely to improve, but for right now you don't need to be coming home to a whole-house clean-up every day, that isn't really fair to you or your family. If he always sleeps with you, you might consider putting a crate by the bed for the time being.
You may end up doing more but less