I just think you should slow down. You pursued H by asking him to reconcile with you, he shot you down. You're pissed and hurt. Wait until you have a better grip on your emotions to act.
I seldom see mediation work. If you want to try it, fine. But YOU should see a L first before you see the mediator. No need to tell H that this is your plan. D is business, you don't show your whole hand. Anyway, you need to see your own L so that you clearly know your rights, know how to protect yourself financially right now, know whether you may kick him out, and know what a good deal is in terms of alimony and child support. Whatever you do, keep any D talk strictly business. Don't try to make a point with it, don't try to make him see the light with it, don't try to prove how great you are, etc... And, the longer the process goes on, the less generous H will be. If you see a good deal that is better than what you'd expect the state to do, snap it up. If you google "new jersey divorce alimony" you'll find tons of resources.
Now, with respect to H: H WILL LIE about the A. Period. People who are cheating, lie. I would be very surprised if X were not having sex with OW. Moreover, given adultery is grounds for divorce, which he probably knows, there is no way he is going to be honest with you about having sex with someone else.
BTW, it sounds like he is getting pressure to spend Christmas Eve with his GF.
Back off, quit pursuing, quit plotting about how to get H into therapy. Until H has more space than he wants for a long period of time, H is not going to reconsider anything. Back off. Take a break.
Next time he tries to D-talk with you, just tell him to back off, you need some space. Slooooooowwwww Dooooowwwwwnnnn. You seem to be brand new here. Nothing is going to happen in a few short weeks.
*Maybe* in several months, after a lot of real space, H will look at himself. Where do you want him to be living at such a point in time?
