I'm of the view that children should receive an education to meet their own individual needs. Sometimes that means children from the same family attending different schools. I have 2 siblings and 1 of them attended both a different primary & a different Grammar to me and my other sibling. It created no ill-feeling whatsoever.
My friend had two children, one at the local state Grammar, the other at a fee paying private school. This was due to her choosing the right settings for each child. No ill feeling between any of them.
It was incredibly irresponsible to be discussing this within earshot of your DD. But I would be sitting her down and explaining that her brother has educational needs that his current school are not fulfilling and therefore you've decided as parents to try a different school, that this is nothing to do with money or fairness, it's to do with educational needs.
Your DM has your son's best interests at heart here, so do the right thing by him when making your decision, and don't just decline it due to "fairness" regarding the cost. It's certainly not fair to let your 7yr old daughter put a stop to her brother getting the education he needs.
She just needs things explaining to her over it, and that may take the course over a number of years, when she's old enough to understand better.
The other option is to wait to see what secondary school he gets into to see if things improve there, however, if he doesn't attend a really good one, things may end up worse for him.
If you do decline the offer, then I'd be making sure he starts attending additional classes in the evening in STEM subjects. He needs to be challenged.
But your DH needs to wise up, your DM hasn't chosen your DS because "he's the special one" 🙄 She's seen a struggling boy and is doing something that will genuinely help him, which is a hell of a lot more than his own father is doing. If anything it's your DH who is likely creating the feeling of resentment in your own household.
I just had a thought, my niece was at a state primary and wasn't being challenged, she was moved into private and excelled. Although she was an only child and money was no issue for them, but my point is that some children are not challenged in state schools, and do better elsewhere.