ConnectingDots, I can only speak with sideways experience - DS10 is quiet (at school!), but of his two best friends, one has a strong streak of "class clown", and the other sounds exactly like MumOfThree's DD above. Over the years, I have watched some teachers go crazy, and others do an excellent job of, shall we say, harnessing it for the power of good?
DS's clowning friend had a terrible year last year, finding himself in a completely quiet class where his behaviour was way off norm. Had he been in DS's class - where moving around and full group debate of big issues was the norm - he would have been great. Class composition and style can make a big difference.
Our non-stop talker friend has been in DS's class for six years running now. The good teachers have found him interesting things to talk about, let the two of them do a lot of group work, and generally found ways to make him talk about important things that were relevant to the class, in ways that weren't disruptive to the group at large. DS and this friend are usually intentionally sat together these days, as the teachers have found ways for it to strengthen the performance of both kids in class. When the friend was younger, teachers would sometimes give him a whiteboard, so that if he had a thought that had to come out *now*, he could write it down immediately, and share at a more appropriate time.
All that to say, maybe you can help the teacher find ways to channel it positively? And perhaps create protected spaces where he knows he will be able to let it out, and ways to divert the thoughts to something constructive in the meantime? (and maybe also recognize just how much of that PG brain is unoccupied by grade 3 art, even the interesting kind?!)