You have sole legal and physical custody and you (and I presume the court? Or just you?) determined earlier that he could not have them overnight. I would leave that as it is, for now; when he says, "I want the kids to stay overnight," just keep it very simple and cool and say that since the current agreement that is in place says the kids don't get overnights, you will keep things that way for the moment. He likely will press and push you over this. I would absolutely get your custody lawyer informed of your ex's change of circumstance, and your ex's new requests for overnights, immediately. You probably will end up back in court and your lawyer should be informed now of what's going on, so the lawyer is ready to move if, or when, your ex goes back to court for more visitation than he is now allowed.
There is no way, really, for you to know if this additional visitation he wants is because he simply wants to show off for the new girlfriend or if, more positively, the girlfriend is a positive influence on him and has him wanting truly to step up and be a more involved father. The latter would actually be a good thing for your kids in the long run. But the earlier refusal to go to doctor appointments is very serious and he needs to demonstrate that he is going to change that kind of attitude permanently. What would satisfy you, in terms of knowing he was going to care for the kids adequately if he resumes getting more time with them? It sounds like you and he may need to see a court-ordered mediator or someone who can hash out a formal agreement about taking kids to doctors, meeting with doctors ( and teachers etc. as needed), and other specifics of keeping the kids' care and schooling consistent no matter which parent they're with).
If the court was not involved in the doctor visit issue before, that's too bad -- if it was all you (and not the court) telling the ex he couldn't have overnights, the court doesn't have any record that dad was refusing necessary interactions with the doctor and potentially setting his child up for harm. Not sure if the court was involved in that issue or not -- can't tell from the post.
Get your lawyer informed but don't say yes to overnights just yet.
If you alone have the ability to decide what's "reasonable" (and the court wasn't involved in formally taking away the overnights over the doctor issue), then be prepared for him to take YOU to court to get a less broad visitation order that will specify exactly what he DOES get. You have to think ahead to what you will do if he does that.
And yes, you need to meet the girlfriend officially. She will be mother to your children's half-sister or half-brother and I guarantee by then dad will be talking about how he needs more visitation so the kids can "get to know their little sibling." Fair enough, but you, yourself, have to know how you plan to handle that when the time comes. Don't let it be a surprise.