"You're saying "no matter what The Caver says, I expect this from you, and you will answer to me on it."
That's EXACTLY what I'm saying! I mean, no offense intended here...how else are they going to learn to respect you if you don't enforce the rules? Right now what they're learning is they don't have to respect you.
And I have a feeling your W is probably not going to be partial to this idea because I suspect she may be using this as an excuse or a crutch to be too tired etc. for you.
So someone may shoot me down on this, but if she says something...I don't know that I'd ask her. Just let her know you have an experiment you'd like to try but you need her cooperation on it for it to really be successful. That way she's involved still and you aren't completely taking over...she's still involved in the process. But I wouldn't ask her...that sounds like you are asking for permission to be a parent to your own kids.