((((hugs))))
I'm so sorry to hear you are experiencing this. I don't have any great advice. But perhaps I can give you encouragement? If your son is like mine, public school is not the answer. Because of his giftedness, he would be miserable.
Homeschooling is hard. Homeschooling a gifted child can feel nearly impossible at times. It is so difficult to find a balance between keeping them challenged without killing their natural curiousity. I was told by a psychologist who works with gifted children that I have to be flexible. I have to be willing to chuck a curriculum when it isn't working. I have to accept that he might burn through a curriculum in months or weeks, so I need to be ready to accelerate him often.
How long have you been homeschooling? Have you maybe not figured out yet what makes him tick? Or if you have been homeschooling since the beginning, could it be time for a change? Sometimes I just back way off on all formal schooling. Well I continue with math, but ease off on the rest of our schedule to allow him some time to just be a kid and explore things that might interest him. It seems to recharge his batteries.
If my ds is objecting a lot to school, I try very hard to evaluate if the objection is from laziness or boredom. More often than not, it's boredom.
Hang in there! I feel your frustration, and I hope you can find a solution that works for the whole family!