May - you are quite brave!!! And you have a logical thought process AND I understand your fears. Your fears of pulling the trigger yourself and then not being able to take it back.
So - ask yourself? What snapped his head BACK to the AP? Fear of loss.
He may have some fears in losing you but he has a of lot more ropes attached to you than AP - kids, shared life, history, financials, etc. In his head these "ropes" will keep you bound no matter what he decides to do with AP. That's his safety net.
I'm just like you. Do I pull the trigger and live with the regret if I had just held out a little longer he would have come around? In pulling the trigger does he suddenly have a WTH moment that I called his bluff and moved on?
I think we both need to drop the rope... let them go to get them back...
And, yes above all else you should adopt the mindset of being the best co-parent ever!!! This doesn't mean you have to be bff's... it does mean you have to be friendly. But you can find kindness and joy in sharing your kids and their accomplishments. It doesn't mean you are allowing the door to be open for him to ask how you are doing... and vice versa. You can maintain an air of mystery and keep all contact related to business items only. Heck, you can even sit next to each other at events if you choose... even an occasion of going out to pizza afterwards, but that doesn't mean you have to let him in to your world outside your kids.
As much as you love your kids don't be afraid to pull back while in the same house. Let him have movie night with them and you go for a walk, go visit a friend, go take a bubble bath... then vice versa - you take the kids to the park without him, etc. You can still do things together but it might help him to see the path that lies ahead should he leave.
May I want you to challenge yourself to do 3 things for yourself each day --- they can be simple!!! Like write in your gratitude journal, take a walk, sleep in, DON'T do the dishes, etc. Make a mental note of 3 things May did for simple joy for May.