How much of his behaviour do you see as a gifted issue and how much is a 7/8 y.o. boy issue? Does he have insight into his behaviour, how disruptive it is, and inappropriate? I ask because some gifted children have a maturity and insight that is beyond their years - and some don't.
It doesn't sound like he self-regulates all that well. It's not uncommon at his age, but at this point, it could start to become an issue with the other students. They may want to focus and work diligently, but find that he's disrupting them. My dd has returned this year to the same gifted class she was in years ago. It's interesting that she and her friends are still complaining about sitting with the same disruptive boys that they avoided working with when they were 9 y.o.
You are probably right in that more physical activity throughout the day will be helpful. Does he recognize the signs when he needs an outlet for his energy. It would be nice if he did, and he could let his teacher know that he needed a couple of minutes to get the wiggles out, so that he could sit down and concentrate again.