Lorna, my understanding of (antibiotic) resistant pouchitis is that it means that the pouchitis doesn’t improve with antibiotic treatment. That might indeed be treated with budesonide, though I’d probably choose to try a biologic medication over long-term steroid use. You have, however, described, I think, the opposite of that: pouchitis that improves with antibiotic treatment, but returns when the antibiotics are discontinued. That’s normally called chronic pouchitis, or more specifically antibiotic-dependent pouchitis. That’s normally treated with long-term antibiotics, rotating among several different ones, if possible. You might need to look for a different doctor, unless you can have the right conversation with this one.
If your backups and pain are associated with thicker stool then there are simpler explanations than prolapse to consider first. Perhaps the thicker stool gets backed up behind a narrowed part of your gut. The key question is whether thinning the stool fixes the problem. Since antibiotics tend to thicken the stool, this can take some patience. If you treat this pain as though it’s a partial blockage (which it may be), you could try varying degrees of things like increasing water intake and limiting solid foods until the stool consistency loosens up. Even soluble fiber (like psyllium) can help maintain a softer stool consistency, though I’d be careful with it in the middle of what might be a blockage.