I think it's more important for your kid to stay in a stable environment than to go to a gifted school program.
I'm not sure what the benefit would really be. Maybe he'd be a little bit more ahead in math in high school.
My daughter took both Calculus 1 and 2 in high school and placed out of it.
BUT.. and this is a huge BUT..
Colleges are now making AP credits a lot less valuable than they used to be.
Let me explain.. My son went to a state school in ohio, I assumed his AP credits would at least mark off some elective slots for him.. But unlike when I went to college, there's no electives left, at least at his school and his major.. He had to take 128 hours (I think that was it), and every class had to count for some requirement.
So while the university gave him transfer credit for his AP classes, they didn't really help him that much. I think less than half of them actually helped him graduate. So the lesson here is , when your kid is in high school, the kid needs to think about what universities he wants to go to, what he wants to major in , and what the requirements are, so he can strategically pick his AP classes.
Hang with me, there's a point at the end of all this.. My daughter went to a private college out of state. They only gave her credit for 2 AP classes period. So she chose Calculus 1 and 2 for the 2 credits to transfer.. But it gets worse.. Since she had a lot of credit for Spanish, but didn't chose to use them for her 2 credits, she had to take harder spanish courses to meet her language requirement. Her roommate was a native speaker of Spanish and even her roommate did not know all the vocabulary words in her classes. So that was completely brutal. My daughter was basically punished for getting AP credit in Spanish. In hindsight, she should have not taken the AP test, so that could have just taken a lower level Spanish class.
So here's the point to all this.. This system is set up now so that what you do in high school can only get you so much "ahead" in your college career. If your son goes to accelerated class now, maybe when he's in high school, he'll be able to take Calc 3 or Linear algebra at a community college (some high schools in Ohio have programs like that).. but is it really going to help him that much? IMO, no. So if it was my kid, I would not pull him out of the school that he's in. It's not really going to help him in the future. I think small kids need the stability more. It's ok that your son breezes through his homework, esp since he doesn't act up. That's great that he's learned to cope with going to class where the pace is slow for him.. That's a good life skill.
Anyhow, if someone else on this thread disagrees, I respect their opinion, but I"m not going to argue. I Wish the best for your son.