Thanks for your post. H has to live with himself if he is lying. I will not take on that burden!
I agree with you about the parenting. You and your H need to talk about it when the kids are not around. Can your H agree to back you up until you 2 can talk privately? That's parenting 101, not to let the kids work the 2 of you and that will happen if he keeps undermining you.
I'm with you all the way. I wish I had better advice. My H and I agree with parenting, that's about the only thing that isn't an effort these days.