I had the sense you were nearing the end of the school year. While generally kids don't stand out as much transitioning schools at the start of the year, in reality kids move schools all the time for all kinds of reasons. If you found a good alternative school I would move anytime. A good school would help you with transition issues if any. We have moved mid year but unfortunately, from a good school to a not so good school. Hard to compare.
Sounds like you have the SPD management well underway. An incompatible school environment is going to cause anxiety and therefore exacerbate sensory issues and so on through the visious cycle as you said.
Doesn't sound like there is a system to work with as such. That can actually be ok if the school is willing and able to accommodate as they aren't tied in to a rigid set of guidelines and rules but if no school accelerates, that can be a problem.
I have no real time for differentiation. Nice idea in theory and we have seen it work once but in an already gifted class. For the most part it is a buzzword and not a reality.
Do talk to your DS. Great if he can articulate the issues but not surprising if he can't. My DD can be very hard to read on the detail of emotional issues even when grossly we can tell there is a problem. Kids really struggle with putting their feelings out there, they often can't identify the problems and they do try to protect us and please us. Follow your instincts.
What school options do you have? You mentioned home school. Can't hurt for a few months. If you homeschooled could you then put him back in a higher grade. How would schools determine what grade to put him in? Would they test or just go on his previous school grade?
Private schools? Any options there.