@GrumpyHoonMain
Before you go in all guns blazing think carefully about what your DH would like in the same situation. If he would also like to meet his partner’s future dps (potential new partner for the residential parent would be more important than you as the kids live with them) then stick to the rules.
Honestly it would get my back up straight away if I was having to 'stick to the rules' that my husband's ex laid out for me.
I have no problem with parents asking this (although I don't buy that it's not to size anyone up and purely to make sure the children are safe because seriously, how would you know anything from a quick meeting that your ex hasn't already in the time he's been with this person), but I do have a problem with people acting like the partner is unreasonable for not wanting to or not 'sticking to the rules' that someone else decided before they were even on the scene.
These partners are people, with their own thoughts and feelings and opinions on things. They aren't unreasonable for that. For some people, meeting an ex wife would be a really nerve wracking experience and uncomfortable. I don't have to put myself through that if my partner, the person who actually knows me, has decided I am a safe person to have around his children. With respect, his opinion on his own kids is the one I'm interested in. I'm not in a relationship with their mother and I don't have to stick to imaginary rules she's agreed without my consent which involve me. I can say no.
As PPs said, if you don't trust your ex you have a bigger problem than not meeting a new girlfriend for 10 minutes on the doorstep.
I agree that it's great if everyone can get along and be friendly but we all know, and your exes partners will all know too, that this often isn't the case and these situations can be volatile. They may not want to open themselves up to that and that choice should be respected.
You can ask by all means but if the answer is no then you don't get to demand or insist. You are no longer with the parent of your other child and as hard as it is, you do lose some control when that happens.