I know I'm late answering this, but I'm also the mom of a 7-year-old son who's "twice exceptional" (gifted but with special needs), so I thought my experience might be relevant.
The very simple answer is that you should sit down with the school and request an IEP for both giftedness and social/sensory processing issues. Be prepared: They will be about 10 million times readier to accommodate the things that are holding him back (the social/sensory stuff) than the things that are pushing him forward (the music, and what sounds like a generally high level of intelligence). This hurts (or it does for me anyway, to the point of almost crying), but it's better if you go in prepared.
And, in this meeting, it's important to be very polite, very articulate, very professional. Wear your interview suit. Tell them right off the bat that your son is capable of preforming at a very high level academically, but there needs to be a clear and consistent plan in place for his social and sensory needs. A line that works well is "Once you've got a plan in place, I will back you up, 100 percent."
If you can swing it financially, it can really help to have a professional advocate in the room with you. Those services do not come cheap, but these meetings are so ungodly overwhelming and intimidating, there really is a need for this.
Really, optimally, the best thing to do would be to have him evaluated by a neurodevelopmental pediatrician and then come to the meeting with an evaluation in hand. However, it can take a long, long time to get this kind of appointment, so I'd recommend having an initial meeting with the school and then a follow-up meeting once you've got a report.
I can't give recommendations with regard to homeschooling since I don't have experience in this area, but I can absolutely see why you'd consider going this route.
It sounds like your son will have a brilliant, extraordinary life in music. Whatever you can do (via homeschooling or the public schools) to keep him going, you can do it.
P.S. Oh -- one of my favorite recent books: Andrew Solomon's Far from the Tree. The chapter on prodigies (and to a lesser extent the one on autism) should likely be very relevant to you.