Dh is a WAHD and we have this problem, too. He's just not with them all the time and really, just doesn't KNOW the rules--ya know? So he doesn't always realize that he's counteracting the work I've done. And when I set him straight, IF he even remembers, he's not nearly as good at it as I am and doesn't know how the kids can try to "play" him... so he winds up TRYING to keep to the rules and getting sucked into them pulling one over on him.
Seems kinda like a no-win situation in our house.

Needless to say, he's struggling to find his own way with them. I'm finding that now that he "gets" the majority of the rules, I have to have a "debriefing" with him daily about what's going on and what I'm trying with the kids or rules that I've implemented that day. It was hard for me to even remember most of the day when we started.
Catching him up to begin with was hard. We had to agree that he wouldn't get defensive if I cut in to walk him through how things were done. So like in the case of the carseat, I would speak up very loudly (and/or take over) and verbally walk him through it: "Daddy, we don't tolerate this kind of behavior. When dd is acting like this, Mommy continues to put her in the carseat and we try again next time if dd doesn't scream." (and either I physically take over or verbally walk them through it). Once the kids understood that I was teaching Daddy the tricks of the trade, they were way more careful.
I will say that THAT whole process of him catching up was NOT pleasant for anyone. But now that we're all caught up, it IS a lot easier. And that only worked because I left dh alone with the kids a few times and he saw first-hand that he had very little ability to manage them and/or they completely disregarded him... and he realized it was not only dangerous, but really... what's the word? "Dismissive"? I guess that's the word. Even then, it was hard for him to tolerate my inserting myself into it (even though we had agreed to it) and he would occasionally come back to "I'm not you--you have to let me find my own way with it").
We still have difficulty with it sometimes, but it's worlds better than it was.