Sleep is a evil thing.

Make sure that you document all of this.

I had a very difficult divorce and it dragged on after I had custody and he visitations.
You can't really win with a sociopath and neither can the child(I am not implying that your ex is one btw, I am speaking from my own life).

Look after your own mental health. Your kids need you to be calm and present. You need you to be calm and present :) So, take good care.

Try taking a sleep inducing anti-allergy medication 12 hours before you want to wake up.
3-4 hrs after taking it, you will drift off slowly but sleep soundly for 8 hours and wake feeling refreshed, energised and mind working at its best.
The downside is that it causes a dry mouth and sinuses.

If you take a half dose you can still get to sleep and might wake in the middle of the night, but if you don't switch on a screen and just let yourself drowse in the dark, sleep will return and you'll still get a good night.

Best not to take it regularly as it can be habit forming. A few nights in a row to establish a better sleep pattern, and after that just once or twice a week when anxiety bites to hard or when you have something on the next day that needs you at your best.

Hard to tell kids things like that, but if you give them the same reason their mother gave, they will at least learn to appreciate and accept honesty and reality as it really is. That's an important aspect of life. Hopefully, their mother will give them good care and love once they get there.