I've been through multiple gastro docs, including my current who is practicing at the University of Arkansas Medical Sciences hospital, which is a highly respected teaching hospital. I've yet to find one that could really help me all that much or could intelligently discuss probiotics and other therapies. The success I've experienced managing my UC has come through my own research and trial and error. IMHO, it doesn't hurt to get a second opinion, no matter how highly esteemed your current GI may be.
My diagnosis has been in question, too... was it UC or Crohn's? I have atypical UC symptoms, but the Prometheus IBD test indicated UC, so who knows? Ultimately, I decided the things that I've found most helpful - probiotics, dietary manipulations, omega-3 supplementation, etc. - are the same regardless of the diagnosis, so I no longer care what tag is applied to me.
I disagree with your doctor's advice to treat the symptoms and assume some inflammation is acceptable. I feel inflammation is not normal and should be addressed, regardless of whether or not you have overt symptoms. There are supplements that can help reduce inflammation and encourage gut healing. Diet also plays a role in controlling inflammation.
I do agree with your doctor that Asacol could potentially be causing problems. Many people have issues with it and, ironically, some of the common complaints are the very things we're trying to improve, like diarrhea. If you don't feel good about stopping your meds, why not ask your doctor for an alternate ASA?