@ HowlerKarma — Thanks for the thoughts. I wonder how much being in a different peer group might help with this.
@ Val — Yeah, lots of gifted children have the issue that you highlight, and what you're doing sounds good.
There's something to watch out for in connection with competitions. As one passes to progressively more elite competitions, the chances of doing well relative to others go way down. I know of children who scored at the 99th percentile in the American Math Competition, and who qualified for the next round (or the round after that) and were among the weakest participants there and felt inferior as a result. So an early boost in self-confidence coming from involvement with competitions can be followed by a drop later on. Some people take losing in stride and don't have this issue, but it's worth being vigilant toward.
