I am glad to hear you are talking more to BH. But you clammed up a few times. I get the sense that you have a hard time expressing yourself and hope things will just get better by autopilot.
I would like to suggest you email him or write him a letter answering his question about "looking for new meat.". And expressing specific remorse and actions you are taking to be a safe partner again. Heck, you can run it by us first.
Sounds like BH is not really opposed to you finding a new job. I am not surprised. Look, it's day care, not an Ivy league school. You can work on finding something better and still be there and supporting your daughter. I know she gets free care there, but I would bet you can find something just as good.
Great news about the MC. If I were you, I would come out and tell the MC that the reason that you are there is because you cheated and you want to save your M. I spent 3-4 M sessions talking about the M problems, and my A never was brought up. Wtf. YOU be the one to bring it up.
That's all for now. Peace.