Have you talked to his sister directly, or only know what he told you? What I'm wondering is, was she not willing to let him stay there because he was seeing OW, and she didn't want to be his "cover" or alibi. She might not want to tell you this, but you can ask.
I want to tell you that you should not allow him to see OW while he is living in your house. But it won't do any good--he has already proven that he will lie and sneak in order to get what he wants. So anything short of house arrest just won't work.
On the other hand, I simply can't advise that you tell him he is free to do as he likes. "Have your cake and eat it too." Your M will never get better in these circumstances, and his actions (flaunting the A) will continue to bring you down.
So is there a 3rd way? First off, I would tell him what you think you deserve in terms of basic respect and common courtesy. Such as, do not talk to OW in front of you, let you know when he will be home, etc. You don't ask where he is going or who he will be with, and he won't lie about it. Also come up with a schedule of solo and joint parenting. That is, you get to go out and do "whatever" on certain nights (your GAL activities) and he is responsible for D5. Maybe even alternating weekends. He needs a heavy dose of 1-on-1 time with D5.
If this idea seems completely unacceptable, then you might as well tell him that he cannot see OW while in your house, and if that is not OK, you expect him to find a new place to live.
There are basically 3 scenarios:
1) He moves out and sees more of OW and less of you and D5. 2) You forbid him to see OW, he resents you and sneaks around anyway. 3) You co-parent and co-habitate. He sees more of you and D5, and with reduced "pressure" perhaps OW seems like less of a necessary escape hatch.
