Sorry to hear things are so tough right now Lonesome. I think the situation you describe is difficult and complicated, and one thing that strikes me is that it might be important to have a discussion with H in MC around boundaries which strike a balance between emotional honesty and speaking considerately to one another.
It really sounds like your H lacked tact, though I can understand from his point of view that you might have expected honesty from him in counselling. I'm wondering if MC might be able to offer him (or you both) any practical skills for managing openness without hurting each other's feelings.
I would also have a conversation with MC (whether with or without H present) regarding what you described here as MC overcompensating and praising H more than you. I think it's important to make your feelings heard about that because MC might not even be aware he's doing it, and it ought to be flagged up before it becomes an unhealthy/negative reenforcement pattern or H starts using it as ammunition (again that could all happen subconsciously).
I just want to reassure you about your daughter - please remember that a good parent is not one that can always cope and always gets things right (no such person exists!) A good parent is one who recognises when they are not coping and seeks appropriate help to minimise the impact to their child. That is something you are very good at, and that is admirable and courageous. It would be a shame if that skill was negatively impacted by H's disappointment and frustration.
I hope you're both able to use this as an opportunity to further constructive communication in the relationship and with MC