He is a deadbeat.
My humble suggestion is to drag him into court and disclose all.
Get as much as you can.

Not a clue.
Not fun.

This does remind me of a teacher who divorced her worthless husband. She made the mistake of not changing bank accounts during/right after the divorce and her x-husband wrote a check for eight thousand dollars on her (joint account) account so he could buy a girl an engagement ring. The teacher wound up paying for the ring as the divorce judge let that slide.

I wish you luck.

Here in the USA, filing for legal costs incurred is just part of it, but very very seldom are the other parties held responsible for anything other than their legal costs. It's just a game - nothing about being fair or reasonable. You ask for everything under the sun, then act with indignation when you have to compromise away things you didn't really care for or expect in the first place. Another reason why I think government has no need to be involved with who should marry whom. It's just a racket for the lawyers/judges to fleece the public.

I wasn't aware that adultery could still be cited as grounds for divorce anymore, it certainly isn't in Canada and hasn't been for decades. People wind up doing shitty things in a divorce, they act out of anger and pain and regret, try to understand that but also hold your ground because you have the good of the children at stake. My wife hooked up with a US military officer and took my kids out of the country, changed their names and refused access in spite of a court order to the contrary and the ones that were hurt the most were the kids. I still send out a check every month even though my kids have long since graduated from university because it is cheaper than going back to family court to get this stopped. Something I dread the thought of dredging up again but still it has to be done now that the kids are all in their 30's.

Gee, I wish I could find the right words for you. I would like to think the law should protect you but, I am not a lawyer. Perhaps that's a good start, find people who is a lawyer or know lawyers so you may get advise. Watch for schamers too, not everyone who may claim is a lawyer necessarily will look for your best interest

I don't understand how the kids factor into this, but If he wants to throw his money away and then blame you for it, I'd throw everything I have at him and watch him squirm.

If he cheated, all he deserves is a swift boot in his ass !

If you could out your assets in someone else's name perhaps - a friend, relative ?

Fight !

Protect your assets at all costs. Your kids don’t need to know the details.

You play with the bull you get the horns. Your kids are going to see the effects anyway whether you tell them about this or not. I'd not play nice after that request. Will you be splitting custody? If not & they're in your charge I would give him what he seems to want, a fight.
Then again, I've never been in a divorce situation. When certain people seem to be pushing me in a precarious circumstance I give them what they want & all that might entail, sideways.

It's about the well being of you and yours now. Don't budge or you'll not forgive yourself later with the coulda' woulda' should haves.

Is it possible to put a legal price on adultery?

I'm tempted to joke about the PlayStation being 'for the players' but I'm sure this is not the time.

I hope you guys can work it out.