I've taken my DC to quite a few festivals and they loved it.
But no I would not trust your DH to look after them.
Going to festival without DC can be a hugely freeing experience, it's great fun.
Going with little ones is a completely different experience. Their needs have to come first a lot of the time, and that definitely includes not falling asleep and leaving them to it.
Which festivals does he plan abandoning taking her too? Please don't say Glastonbury! (One of my favourite festivals, but no place for a child if their parent isn't really on it!)
The best way to take DC to festivals is in a big bunch of you, with other DC. Then you can take turns in being the responsible one who has to look after the DC, and people who want to drink (and fall asleep) can do that at some point - but again only if they're on it and not as you say just assumign someone else has the kids at all times.
Taking a child on your own can be hard work as an adult, and not so much fun!
If this was the only issue I'd be tempted to try to compromise and go to a festival you think you have some chance of enjoying, and inviting friends to as well.
But then the lying takes it to another level.
The asking her to lie to you and not taking it seriously are not on.
He knows he has a problem if he's lying to you about it, and getting his DD to lie too.
Not great. 