It does sound like enough time has passed for things to be more civilized. It is possible that she has issues, and would get upset seeing her ex with another woman, but the arrangement you've described doesn't sound like a healthy setup in the long run.
If nothing else, it'll be confusing for the kids, as you're going to have to be part of their lives in some way....
I would suggest speaking to your partner to get to the bottom of this. Don't feel bad for bringing it up, it's your right!