If she were to agree to go a counseling session, I think it would be to tell you she wants a divorce, not to admit to an affair. I'm not saying admitting to an affair couldn't happen, but the likelihood is slim.
If you feel you will always look back and regret not attending at least one session with her, then do what you have to do. I am simply trying to prepare you for a different outcome than your heart is seeking. She has no well wishes, no desires for the betterment of your MR. Her goal is to end the MR. The whole pretense of learning better communication for the sake of your kids, is a scam. I think it's her way of transitioning the family into D. Many so-called MC, are really D counselors.
She can tell her friends/family that she even tried MC, and nothing worked. Do you remember me saying that a WW is motivated by selfishness? If she agrees to something, it is b/c she has something to gain. She's not interested in piecing the M back together, so she would be going for other reasons.........like stated above. She can do all of this without ever admitting to an affair. Observers think she has "tried", and she walks away feeling a sense of justification of attending MC.
IMHO, I think the best plan is to stop playing family and marriage. Let her go. Allow her to experience life with out you. Get your lawyer's legal advice about the kids, bills, accounts, financial support, etc. Check to see if it hurts the H's with child custody and finances if he should have to temporarily leave the home (if she refuses).
I believe the H has to be suited in full armor b/c his W is not going to play fair, so he must protect, protect, protect. No more pursuit type actions in saving the M (which usually means to him persuading his W). He lets go of her, removes all emotional pressure he was putting on her to reconcile, puts physical distance between them, goes no contact, and let's her see life without him. It throws her and the affair partner closer in some cases, but it also has been known to end the thrill of their secrectcy and a breakup follows pretty soon.
During this period, the H stays dark; GAL like there is no tomorrow; focuses on self healing rather than relationship healings; does not keep tabs on his WW's life; no social media stalking/snooping or other self defeating avenues. When her fantasy begins to crumble down around her, he does not become her rescuer or street cleaner. He makes no attempt to reach out to her. He is busy healing and making a life without her. This is her mess, and she'll have to clean it up without your assistance.
No doubt, she will notice her H's lack of availability to her. Along with the loss of his availability, accommodation, and attention.......may come other uncomfortable experiences in her life. If so, the H needs to allow her to experience it, without him rushing in to save the day. If her affair ends, and if she doesn't get involved in another OM.........she might possible start breaking through the fog and thinking more rationally. At any rate, any initial contact should come from her. I don't mean her reaching out wanting rescued, wanting you to do something for her, or anything selfish whatsoever. If it something for "her"........then let it go, b/c she is not ready to change. She just wants to use you.
H's ask how they could dertermine if their W was genuine about wanting to reconcile. She would need to make the first move to stop the D, or apologize to you, or ask for another chance, etc. If she feels remorse, she will be very humble in spirit, tone of voice, attitude, body language, approach, behavior.......everything. There would be no spitefulness, resentment, anger, blaming, or arrogance. The dynamics would be reversed. If you can't see any of these things, then she has not done the work a WW needs to do to find her way back. Don't be fooled by her tears. If she has worked on her heart, you should be able to easily detect signs of humility.
Getting back to dynamics in a MR. Here is something to remember. The spouse who cares the lest about the R, is the one who holds the most power. The fearful spouse who frets over what the other one is doing, or not doing, or may do, has zero power in the R. The spouse who worries what the other spouse feels/thinks, likes/hates, what response to expect, always walking on egg shells to keep the other spouse in a better mood, etc., is the spouse who has no power. Currently, you are the spouse with no power in the relationship. You won't get it by pursuing her.
I have been known, occasionally, to tell a LBH to stop trying to save his MR. B/c his mindset is so desperate that most of his actions are pathetic and repulsive in the eyes of his WW. He gets into a frenzy of beating the divorce clock. He starts grabbing at any straw he sees. IMHO, that is where you are currently.
I may be wrong, but I doubt an affair would be a deal breaker for you. At the moment, you seem focused on evidence of an A, but if you had a video of them having sex.......I still see you trying to save the M by adding that if she'd agree to MC, the M could be saved. The affair is her acting out in rebellion. The real problem is that she stopped feeling respect/admiration for you as a man. That killed her loving feelings. It can be turned around, but it is not for the unwilling. Currently, her heart is unwilling. Her heart is closed to you, and it only feels a ton of negative feelings for you. Everything you try, feels like emotional pressure to her. Until she feels free of that pressure you are applying.......I don't see her making positive changes anytime soon.
I'm sorry if you see this post as doom & gloom. I hope you will see how you need to take a different direction. Let her go. Stop trying to hang on to her. Stop trying to make her see that the MR can be saved. Stop telling her you don't want a D. Stop trying to make her feel a certain way, or think a certain way. Just let it go. You've been in a tug of war with her, .......now, drop the rope, turn and walk away. She'll probably fall on her face, but you are not going to catch her. Let her fall.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!