It sounds like school may be difficult to work with as they've taken quite a stance. I did get more from our school when I showed test results we had obtained privately. If you can afford the testing, that might help. Cost varies among areas but an IQ and achievement test can go a long way to illustrate what the problem is and how severe it is. Lots of good reading at
www.hoagiesgifted.com if you haven't been there yet. I recommend reading about levels of gifted and about testing often used.
I have a DS who is about to turn 7 and we have just made the transition to home schooling. Surprisingly it is so much easier than I thought. Learning has never been our problem but teaching methods and insistence on keeping him with his young agemates has been toxic.
You are in a great place here on this board to explore all your questions and options. The wisdom and experience represented here is staggering. I hope you soon began to have more solutions than problems.