@Petunia1970 I have moved in with my father and gave up work so in many ways it's less stressful than having to juggle things but at the same time very claustrophobic.. it helped that we started it during lockdown and before dementia kicked in.
Good luck with Social Services. I think it is also time to bite the bullet with your sister and see if she can give you respite. It might be worth doing a self-refferal for your own mental health via the GP.
In a sense though it is your Mother's house and she sees that the boundaries are being crossed which is what's probably upsetting her. It may be that you have to ask which is annoying as it's a loop conversation or see if would allow someone else to do it. It's taken a very long time for my father to accept that I take a lot of the decisions and he hates the lack of non-consultation. I still get it's my house but at least I also get instructions on what I should do about a problem.
Ultimately the decision is how much guilt can you live with. Not being there and worried or being there and being unable to do everything you feel you ought.
Sadly there are no easy solutions and everyone is very different.
Good luck.