It is easy OP, he can find himself an Airbnb, a room somewhere, whatever, a grown adult man who works can find something; people do it every day!
You don't have to disrupt them, you do want to disrupt him though after him fucking up your life, least he can do is go and give you space; but they never do, they just put on the tears and ensure they keep their comfy lifestyle at home.
They have every reason to lie actually.
You continue to make excuses for him, you were not over, you were very much in a relationship; it's all just excuses, fact is he's a cheat, I'd put money on it being sexual, not many men would risk their lifestyle for less.
Sorry OP, but you are completely in denial, understandable though but honestly, why would they not have had sex, or some kind of intimacy, their late night calls were probably sexual in nature, they could quite easily have had explicit chats online without actually meeting face to face.
As i keep saying, no matter how stale or shit things had got it does not give anyone the right to cheat, they cheat because they have no respect or value for their relationship, simple as that, and he won't change that opinion, he's showed you it loud and clear.
Brush it under the carpet, let him carry on without any consequence, be prepared for it to happen again is all i can say.
Has he offered to go to counselling to try and work out why he thinks it's ok to cheat on you and give you actual answers, nah, didn't think so.