As you can tell, you are WAY too much fun that I'm ignoring my stomach bug to post here today. [can you tell that I'm not actively working here]
If I were you, I wouldn't talk with H based on his past track record in spewing, twisting, and what not at you.
I'd approach the L about the legal aspect and try to set up a family counseling with the psychologist. Get all of your ducks in a row FIRST before they are presented to H.
The most ideal way would be for the child psychologist to open up the first session in addressing co-parenting between the pair of you. If I recall correctly, Bets had something similar set up for her family and it was when Mr. Wonderful begun to see the effects of HIS choice. Then Mr. Wonderful stepped up the place as a Dad. We can only hope that your H does the same.
Only one way to find out...call the child psychologist for her input.
I'm sure Bets, Bug, and 25 will all chime in for some additional suggestions in due course. I'm gonna bow out of this one and let the vets with kids guide you here.