I’m not sure, so if I were you I’d test the waters before committing to moving in.

Have your girlfriend over more often, stay for dinner, those things. If she’s already doing those things or it goes well, have her stay the night a few times. After that, have her stay for a week. Pay attention to how the kids and your ex react, not just what they say about it. Do they avoid going into rooms where your girlfriend is? Do they seem more annoyed than usual at signs of her presence, like a left-out plate?

When you do all this, treat her like a resident, not a guest.

If all that goes smoothly, I’d give the move-in a shot. If it doesn’t, then you haven’t committed your girlfriend to giving up her current living arrangements, she can go back to them.