I don't think you can answer this question without an indepth look at what resources you have available as a homeschooler.
Do you make friends easily?
Will you be able to find good opportunities for him?
What is the homeschooling community like in your area?
Will he be in co-ops or clubs?
Are you good at coaching him to try new things?
It is very important how it is presented. I can see it would be very possible for a kid to think they failed and they weren't allowed to go to school any more. So, I would try to bring him in on the discussion. That doesn't mean he's the one responsible for making the decision. But, it does mean that he should understand the choices aren't school or life is just like it normally is during the summer. He should understand that he will still be in activities with other kids. He should understand you will still have expectations that he try new things. I would present it as a chance to work on other things. Also, if you really think it would be better, personally I would not be above manipulating a bit by finding ways to make that first year of homeschooling pretty special and exciting.
Finally, for what it is worth, the level of anxiety we saw at 7 is entirely different than what we saw at 10. Maturity, more experience, practice, CBT, all can make a huge difference. I understand the feeling that any progress he has is fragile, but I would remember all of life offers opportunities to learn to deal with anxiety. Public school doesn't hold a monopoly on that.
