Sometimes you just have to get tough! He’s using your niceness against you to get what he wants, and has guilted you into being his hostage rather than a loved partner. I get that you care for him, but his illness is his responsibility. If he threatens to kill himself, you can call the police - he will either get the help he needs or suffer the consequences of trying to manipulate you into staying with him. You’re not responsible for the choices he makes!
And who knows how much longer the pandemic is going to last? I agree, he doesn’t necessarily need to leave immediately, but agree on a deadline if you can, and make a plan for that time, including what you will do if he starts refusing to leave etc.

My last ex threatened to start harming himself again when I was breaking up with him (I had also been delaying the breakup because he’d told me how depressed etc he’d been after his last girlfriend left him... red flag I didn’t see for what it was at the time...). I still broke up with him. I’m pretty sure he didn’t harm himself. He seemed ok, just tired the next time I saw him. It was basically an attempted guilt trip to try and stop me leaving him.