Options to protect Mobile Phone from over charging which is causing battery failure
We have a number of Android mobile phones permanently on charge.
  Note that the screens are on permanently at full brightness.


Phones are not intended for such high duty cycle operation. They will overheat. This will cause the battery to be damaged.


  Would this strategy of trying to balance the charging current of the USB charger with the power use of the phone be effective at preventing over charging?


No. In fact it would make it refuse external power at all. Working from battery alone due to the voltage drop.

Options:   


Buy the correct devices. Search for HMI panels with Android.
Remove the battery and connect to the battery terminals directly.
If the phone is picky, pull the management board from the battery and use those BMS battery pins to connect approximately a 4V source, simulating the battery. Success not guaranteed. Warranty most definitely void.
Do this on a discharged battery and don't puncture or short the pouch.