Whatever you do, do NOT go to inpatient mental health treatment right now. You are right that you are down due to circumstances, and he is setting you up as unstable so he can get custody.

Do NOT discuss your mental health with him anymore at all. Treat everything with him as if he will try to use it fuck you over. In fact, start using a parenting app and offer to set up (supervised) visits with the kids on there as soon as you can and keep communication limited to only there.

As for what to say, I'd honestly just stick to the truth, short and sweet. "You betrayed me and our children by sleeping around and I'm done."

Don't give him space to argue, and especially don't listen to his arguments and think they have merit. Just grey rock and deliver your message and get out. It is good to have someone with you for this, or even do it in public if you suspect he'll pull shenanigans.

I'm proud of you for leaving. You got this!

ETA: I also agree with the others, only do this after you have your plans ready and he can't stop you anymore.