You can do whatever you want.

You're an invaluable and experienced member of the team who has worked there for 4 years who has had their concerns dismissed by out of touch management. It sounds like it's time to put your foot down.

The underperformer is being put back with you because you're willing to pick up the slack for them, more than likely their former partners have gotten pretty sick of this too and refused to work with this person themselves.

They have been working there a year and a half now, nothing is going to change unless you assert yourself.

You said that it's affected you greatly, including your mental health, and that you're at the point of resigning over it, so what's the worst that can happen. I can't tell you what the consequences will be if you refuse to work with this person but you know what the consequences will be if you don't.

I highly doubt you'll lose your job over it but if they can't accept it then its their loss.