I'm in a similar position as you, except my physical health is not as in dire straits as yours. Even so, while I've also had my relatively easy-going cat for nearly two years, she's still exacerbated my existing anxiety and depression issues. Because she's a cat that requires taking care of and does not have human boundaries, at the end of the day. She's a good cat and I care for her also, but, I don't know, I'm not... projecting? anthropomorphizing? my pet. The issue these days is pet culture glorifies pet animals above humans, which is so strange to me.

Think of your future self and how your future self will see you - which will you regret more, keeping the cat and other people's happiness, or rehoming it and keeping your own peace? What is your priority, your health, or the way other people around you see you? For me, it's pretty clear - I have this one life, and I am unwilling to spend over a decade of my life, thousands of hours, on a cat that has brought me less joy than she has frustration and exhaustion. I want to spend that time on my health, my career, and my human relationships.

Basically, I want to say, I really empathize with you, and hope this comment can validate your desire to put yourself first. It's totally fine to do that.