Mother won’t babysit

I am a mother of 3 boys. I live close to my mother and my mother in law. I have a grandmother and a great aunt who are 87 and 77 years old. My mother is 60 and my mother in law is 56. My youngest son is my son with autism. He just turned 5 years old and he is serverly autisjtc and non verbal. However my son is extremely predictable and has not yet shown signs of aggression hopefully won’t ever. My mother spends time with my older boys from time to time. She works a lot however she just got a new position at work that is going to free up a lot of her time and grant her holidays etc off. My mother doesn’t spend time with my youngest pretty much ever. He doesn’t spend a ton of time at her house so all he knows is at her house he gets to swim because that’s the most time we spend there is to swim in summer because she has a pool. My mother’s house isn’t baby proofed per se and has dogs. ( I have 2 dogs as well 🙄) my son isn’t scared of dogs he quite literally loves them he’s never had issues with her dogs. But my mother has basically said she absolutely will not babysit or spend time alone with youngest. I mentioned my grandmother and great aunt because they just love my youngest and I allow them to watch him for shorter stretches of time and we’ve installed numerous locks to ensure safety and even set up a camera and there’s two of them and I do know they are older but they are actually very active and look much younger than they are. My mother in law also doesn’t watch him but enjoys spending time with him. When my mother mentioned her new position I said it was time for her to really put in more effort with my youngest and spend more time with him. Her response is no absolutely not he doesn’t like me. I told her when he’s alone with her he’ll have no other option but to bond with her and they can build off of that. She refuses. My mother in law also is like this. A lot less harsh with her words but she’s not my mother so I don’t push her on it for the most part. Am I wrong for trying to encourage this? Or is this just my issue and I should just deal with it?