My relationship was in a shit place for quite a while after an incident similar to this.
I did also finish it for a while.
(No kids involved)
As I said I went and did the same or worse to him and he could say nothing about it
We both came to the conclusion that strip clubs are not appropriate places for people in serious relationships to be, and that "just sitting in them" often involves more (even if someone doesn't purposefully buy a private "dance" like your h).
He knew if he entered one again, our relationship would end
Op, if you stay with him, it's entirely up to you of course but I'd make sure he feels some reciprocal pain & discomfort on this front. It's not easy to find male lap dancing clubs but some floor shows/stage shows will do "dances" ..... Though still not like a private dance but ..
You can also just hire a stripper for you/your mates on a night out/some will come to your accommodation.
Men need to feel what this is like so they can stop their BS, and he needs to feel some pain for this so he thinks twice very hard about repeating similar.
I still think he's got shit integrity and judgement though.