Rali, you've got this! Share your concern with your H. Share what 6 yo said after telling him you know you can trust him to do right by your D6. And that her complaints have NO bearing whatsoever on her time with him. That you can't and won't be two parents. That D6 needs a good man to be her father. And that you married that man.
Honestly I've waffled back and forth on this for the last day or two. While STBX used to be a good guy who put his kids well being first, I think any reasonable person would look at his choices lately and agree that is not the case right now. Truthfully, I'm not sure this would be a productive conversation and I can see him easily slipping into super defensive mode.
I think I might approach this in co parent counseling. While he's been awfully civil the last few days, it doesn't take much reading on these boards to see how easily that can tip into something else - and we probably need a third party to at least get the ball started on this topic. After all, the root of my issues with co-parenting are that I cannot believe anything he says right now, and somehow we will have to build trust up again. From where I sit today - that is hard to envision.