Hi yohoohoo
My first husband was a cheat. So, I hope I can give you some advice here. In my case, I didn't know anything for sure, but during our 20 year relationship, I had lots of occasions where I felt something wasn't right. I'd question him. He'd always come up with a convincing explanation. I couldn't act just on gut instinct alone, as I could never find proof. In the end a friend told me the truth and the whole sorry tale came to light - over 10 women (and I'm sure many more I didn't find out about).
There are lots of things that bother me here :
He says that she approached him...not sure I believe that, but even so, giving him the benefit of the doubt, his response when she asked for his number, was to give it, when he should have said "sorry, but I'm married".
He says he met her that night. So, he was willing to risk his marriage and family life for a stranger? It's not someone he was close to at work, and unintentionally started to develop feelings for. He is literally willing to lose you, for someone he doesn't have any feelings for. It's purely a sexual attraction. So, a quick shag and some flattery?
You don't "miss" someone, unless you were with them before "the missing".
Why would grown adults, who swap numbers and snog, not take it further that night? Presumably he had a hotel room. And she probably lived in the town that the casino was in. They're randy and have had alcohol. Not sure I buy that.
If you don't (at the very least) make him leave for a while, he will think he has got away with this. So, what do you think will happen when he is next in a bar, and gets chatting to a pretty lady? Behave the same, imo. But cover his tracks better.
You'll never get to the bottom of this, but imo, what you already know makes this a deal breaker. You may not LTB today or next week, or next month. It's very hard to do (it took me 4 years). But one day it will all come to a head, because from now on you're going to be a paranoid wreck every time he goes out, and it will wear you down.
As an aside, is the OW married? If she was having an affair (emotional or otherwise) with your DH, knowing about you and your DD, then I'm afraid I would want to throw a hand grenade in to her life by telling her DH, just exactly what she has been up to.