A not knowing how to handle the lack of integrity in the co parent

In this post you focus on her parenting and how it makes your role as an educator difficult. I definitly understand how this would be difficult. I also understand why you have singled out her as the problem and why you prefer to keep focus at that.

But the reason why this doesn't work is not hers alone. It is because their mutual relationship has not worked since they brought kids into it.

What stood out to me when I read your post was .. has this guy completely abandoned his obligations as a father? Has he abandoned his obligations as a husband? He is definitly not standing up for his kids, and my educated guess is that he never rose to the challenge as a father and husband. (In my personal experience this is very often the case with dom-types, but that's beside the point).

They don't work together as a parental unit in this family. They don't work together as a couple. When a relationship struggles it is always issues on both sides, even when you onlly see issues on her side. The only exception to this rule is when you are dealing with an abusive relationship. So the problem is the dynamic that they have created together.

You are trying to get around this by explaining his role as the main provider and hers as the main caregiver, but the old days are gone. He is neglecting his parental duty, in practice "abandoning" his kids, and I highly suspect that if you looked under the surface of their relationship you'd likely discover that he abandoned her as well at some point on their journey.

So you entered this closed dysfunctional system, not as a paid professional, but as his lover and sub, and her meta. And now their relationship is melting. This always happens when you throw a triad poly into something that doesn't work in the first place. It is not your fault but you may have to make a decision.

They need couple's counseling and you need to move out in order to protect yourself. You can still have a relationship with him but not in his/ their house, not as a lover living in their house.