Hi OP.
I can see why you're so so gutted.
The hideous thing about a situation like this is that it can't ever really be made right, can it? What you want - what you thought you had, and what you committed to - isn't a man who agrees not to go to strip clubs, but a man who wouldn't want to go to strip clubs. Fundamentally different.
That's what I would want, that's what I believe I have. A man who would think 'Oh my God, that's foul' at the prospect - for all sorts of reasons - the issue of exploitation, because he would see engaging at that level with another person as infidelity, because the idea is tacky and shit beyond belief.
The fact that he thinks that way, for all those reasons, is part of WHY I am with him. It illustrates his basic belief system, his morality, FGS his basic dignity.
What you've discovered is that the man you thought you knew is actually a very different person with a very different belief system. That is fundamental.
And that means that essentially, all this stuff about winning back trust, about 'working' to make it right again, demonstrating through actions etc. etc. is missing the point. He hasn't made a mistake in that sense. Objectively, he has done nothing wrong in going to the club - it isn't illegal. It simply has shown you that he is a different person to the one you thought. And you have a problem with that. So, he could promise never to go again and keep that promise, and it wouldn't change the fact that you do not wish, would not have chosen a relationship with a man who thinks it's ok to go to strip clubs and as a result the relationship has been spoiled for you.
THAT is what has you banging your head against a brick wall. Because 'getting over it' essentially means compromising your principles and staying with a man who is less than you thought, committing to children with him.
It really is as big as that.
And I haven't even mentioned the lying about it yet... Lying like that, about something real and important (hell, lying full stop really) would be a dealbreaker for me.
Put the ability to lie alongside the fact that you now know where his moral compass lies, and you have a recipe for future infidelity.
I don't know what you should do. I know two things you shouldn't do, one is to try and bury it, the other is to stay with him because you're afraid you won't have children if you do.
Good luck.