What a revolting man.
He cheats, then he thinks he can keep you both on the go. After telling you that he'd miss her more than you.
How entitled, utterly selfish, vile, arrogant.... I'm amazed.
And yet you're trying to support him through it? What??? Support him through the effects of cheating on you and treating you like dirt?
He has no right to choose what happens next. Why would you want him to 'pick' you? Even if he did, you would always know what he's done, and that he went out looking for someone else, and he cheated, and lied, and didn't think of your feelings. You wouldn't ever trust him again.
If you do nothing, he will likely mope about for lots longer yet, keeping you on tenterhooks while he decides which of you is the lucky winner of this prime specimen of a man (not).
If he chooses her, you will be further hurt and have wasted even more time.
If he chooses you, you'll never trust him again and you relationship will never be the same. He'll likely cheat again.
Please stat taking steps to separate. Don't ask him to stay with you. Don't let him think you'll accept being treated like dirt. And make sure he's absolutely clear that he wrecked your marriage. It is enrirely his fault. And don't let him blame his MH. Being mentally ill doesn't mean you have to treat other people like dirt. I've never heard that being mentally ill means you need a woman other than your wife to give you blow jobs.
His MH isn't to blame (and even if it was, he's refusing to get help for it, so it's still his fault).
He is to blame. Completely. Don't let him play the victim. And if he threatens suicide, advise him to see his GP and you can call the Police so they can do a welfare check. He is using that threat to manipulate you and to add to his 'victim' status. It is not genuine - and even if it was, you aren't responsible for him. You can't stop him killing himself if he wants to. If he is so down that he is suicidal then he needs to seek help from professionals (and stop doing dick-ish things that wreak havoc in his own life).
Please look after you and DC. That's your only responsibilty in all this. That man is only thinking of himself. (It's interesting that you've always been supportive, and he expects it, but when you needed help he was spending time calling her).
Best of luck.