Just in case you're still thinking of ideas to get the bill, I would suggest the fax idea someone mentioned before. Fax them a request. And yes, it's lying, but use his name on it.
My FIL died last November, and we've been learning LOTS of sneaky tricks like that, in cleaning up the utter chaos he caused. He took almost all of his wife's life insurance by faxing in requests for info and then ultimately the request to take that money out, and forging her signature (we even found the practice sheet he used). He died, she went to take all that money out herself, to live on (he had no life insurance b/c they thought she would die first b/c of two heart attacks and diabetes), and it was almost all gone. Also found out about a 7K loan he'd taken on the insurance policy 10 years ago, and never paid back...
sneaky tricks...
But really, I think you know the answer. Counseling is good and all, but I, personally, don't believe in joint counseling until the individual problems start getting ironed out. A person doesn't cheat b/c of problems with the other person, they cheat b/c there's something wrong inside of them. And you can't work out your OWN stuff when there's someone else in the room, having their own problems. My DH never cheated, but he had some problems, and he had individual counseling for MONTHS, before asking me to start going to a co-counseling session. And even after that, we used a different counselor, so there was no chance of the counselor taking "sides".
My friend's now-ex husband, well, a cell phone was their downfall too. They were at a NY Eve party and she hugged him, and felt osmething in his suit pocket. Took it out (thinking she had complete right to take things from his pockets) and it was a snazzy and tiny cellphone. That was the start of the end. It wasn't just a new phone on THEIR plan, but a plan all his own...he was using it to call another woman, etc. They gave it a good shot, even moved back in at one point, but ultimately his issues were too much for even him to deal with, and they divorced.
Back to lessons learned from my sneaky lying FIL. Your husband might also have his own mailbox at a place like the UPS store. That's what FIL did. We're still not sure if there are more out there, but that's the one we found.
Things were easy-ish for US to find out, once we went through his files, b/c he kept records. He had his wife trained to keep OUT of his stuff or there would be Hades to pay, and he never encouraged her to read English (she is Korean), so even if she had poked around in his files, she wouldn'lt have been able to understand all that much anyway...
Assuming both of you read English and you're not as beaten as MIL was, your husband probably didn't keep his records like that, and it will be harder for you, if you continued looking.
I hope he can get individual counseling and work out his problems, I hope co-counseling goes well (try to make sure no one lies to the counselor like my friend's ex husband did), I hope it works out very well.
But if it goes the other way, be strong. And don't let anything your kids do or say freak you out too much. When my mom finally dumped my dad, I was 4 (and had had 4 years of watching what he did to her), and I became a total daddy's girl. Was ANGRY at my mom, stopped calling her "mom" and went to her first name. (luckily she was a hippie and she really didn't care about that, so it backfired

, and I called her by her first name all her life, and let my son call me anything he wants to!) She realized that I just wanted the fantasy of two good parents, even though I never had that, and she just rolled with whatever came out of my mouth, and tried really really hard to never take it personally. So if things go that way, I hope you can be that wonderful for your kids, too!