When you’re on a center/app frequency, the radio exchange when an AI flight wants to descend is always:
ATC: <callsign>, report ready for descent.
Plane: Ready for descent, <callsign>.
ATC: <callsign>, descend to…
This feels a bit canned and also isn’t too realistic, since most of the time it will be the pilots who request iniciating the descent. I suggest just having the flights initiate the request with “<callsign>, requesting descent”, it would also de-clutter the frequency a bit by having one less call for each such exchange.
It’s still helpful to have the ATC ask you for descent for your own flight once you reached the TOD calculated by BATC though, in case you were distracted.