I took him to a vet who is supposedly good with figuring out root causes of animal difficulties. She was pretty good to Pico, and he was exceptionally well-behaved, so the exam went very well. We found out from the vet that he's quite healthy, except that he has lost about two pounds since September - not good, but likely a side-effect from all of the hairballing he's been doing. We had a full blood-chemistry panel run to find out any chemical answers, and he got a 2-week antibiotic shot to take care of the raw tummy issues.
The answer that came back from the chemistry panel was, sort-of-surprisingly, not hyperthyroidism. Pico either has anxiety that is causing him to irritate his skin by licking too much, or allergies which cause anxiety. So I took him back to the vet yesterday for a steroid shot. If the problem is allergies first, then this shot should show immediate improvement in Pico stopping his compulsive licking, and we will start him on allergy meds in three weeks. If it's anxiety, then the shot won't do much good and we will start him on the same meds his brother is on, in about three weeks.