I totally agree. However - I have had many issues on which I fundamentally disagreed with my H and tried to work on with him to fix the harm I believed he was causing. I have learned through counseling to try to have conversations instead where I own my beliefs and accept the limitations that they are my beliefs and not necessarily universal truths, and I ask him questions about what he believes to find out where he's coming from.
I used to fear having these conversations because the way I came across was that I knew I was right and he was wrong, and knew he was going to get mad or think I was trying to control him, and/or did not know if I was going to be successful or not in persuading him to the "correct" point of view. All wrong.
He has as much belief that he's doing the right thing. You might start by thinking about what he's trying to accomplish. Tell him you have reason to believe D's being asked to keep secrets (if you can do that without tattling on her that would enable her to continue telling you things if this continues to happen). Ask him why he thinks she should not be able to tell you anything? Is he trying to avoid getting you mad or avoid having to deal with you freaking out on him? Can you reassure him that you want to be able to discuss this with him calmly?
Does he disagree with you about whether it's right to be intimate with a lover in front of her? Would he be OK with you doing the same? Tell him you want to be open to the truth so he can share what he's doing and not keep it a secret - you may not like it but you would appreciate knowing it anyway. Try to open up the communication between you two because you're going to have to have a relationship coparenting and need to be able to talk about hard things.
For D, I would recommend telling her she did the right thing and to keep telling you anything she wants. You're the parent and you can make sure it's OK for her to tell you anything. It's really too much for a little kid to cope with having to "take care of" her parents - help daddy not get in trouble, help mommy not be sad. Take that off her shoulders. We taught our kids from age tiny that if anyone ever told them not to tell their parents something, that they should say OK and then immediately tell their parents. It was the only time we told them it was ever ok to lie. We said it would always be OK to tell us a secret.
Good luck to you, will be thinking of you.
Adinva 51, S20, S18 M24 total 6/15/11-12/1/12 From IDLY to H moving out 9/15/15-3/7/17 From negotiating SA to final D at age 50 5/8/17-now: New relationship with an old friend __ Happiness is a warm puppy.