honestly, I wouldn't take the girls to that house. Chances are that with a start this rapid, they won't be together long. Keep the girls with you, and as long as you aren't violating court orders or changing your plans, he can see them when he comes over. No way would I play his little "I'm in love" games and show the girls that crazy dysfunction.
It will just upset them to see him doting on some other woman's kid. If it lasts 6 months or so, then I might consider it. But not now.