OP I really feel for you.
As the ‘responsible’ child, I have dealt with this all my life, from supporting said parent, to running things in their absence, to supporting them through being institutionalised and being the target of their rage when things got too much.
It got to a point where their MH issues dominated and they were paranoid and nothing I could say or do impacted or helped and our relationship deteriorated to the point where I had to make the decision to step away for my own sanity.
I finally said to myself, why is their wellbeing ahead of my own, and the answer was, it shouldn’t be, I had other siblings (as do you!) to take the strain for a while so I could put myself back together.
In this period they hit rock bottom and had intervention from professional services and they are now on medication and can cope with life much better. Our relationship is now much improved, on occasion when things arise now, I find it much easier to step back and let things play out without intervening too much (or if at all!).
I am also a parent and prioritising me so I can prioritise DC is of utmost importance. Be kind to yourself and do the right thing by you and your family, it sounds like you have done the heavy lifting for a long time and it’s time for someone to step in and take the reins for a while (your Mum, your sibling, MH services). Stop sacrificing your happiness and wellbeing for others and start thinking about what your and your DC need to have a happy life. Don’t feel bad about it, it gets easier with time, the hardest bit is stepping back.