Hi :)
Firstly, I expect putting it here rather than in AIBU was sensible!
It sounds like you are having a nightmare. I am glad for you that you have managed to open DP's eyes to his unhealthy situation. As you say he may have lapses, and you may have tricky times ahead.
This particular question is very difficult isn't it? And probably needs looking at from all sides. Also there probably isn't a 'right' answer, though, given the situation with DP's ex, it is may be better to stick to your guns as you are suggesting, then you can relax things as she adjusts to the new world, and learns the new rules of engagement!
On the other hand, her world has changed significantly and it may not be possible to change it completely. Would the best thing be, as you suggest, to say she can come in and see the rooms another time, but 'not today, thanks.' ?
However, this is your home as well as DPs and I think you are absolutely within your rights to keep her out completely if you so choose. Why exactly should be invited into your house if DP isn't invited into hers?
So, I realise I've given three different answers, mostly so I could consider them together. For me, on balance, I think I'm going with the last one! What's sauce for the goose is, after all, sauce for the gander. And most certainly, every time DP gives in to her, she will expect him to give in to her the next time.
Can I just say also, to demostrate that I am looking at it with a wide view point, I have little problem with having DH's exW in our house (though she isn't my favourite person in the world for various reasons), my exH is not allowed in our house for very good legal reasons, and I have never met exH's wife or been in their house - nor am I traumatised by not being able to visualise where my DCs sleep when they are there.
And it's up to DP and you what the sleeping arrangements are regarding sharing, not her.
And no, you absolutely do not have to meet her alone/for coffee. You, in fact, don't have to meet her at all if you don't feel comfortable.
I know that's a really long answer, and not terribly conclusive. TBH I rather feel for your situation and wanted to give you an answer of sorts, even if it is just to say that nothing you are suggesting appears unreasonable to me :)