Jeff - No doubt you have a very odd problem here.
I was thinking what could cause such a strange problem. Then it hit me!
Your charger supplies the needed power to run and charge your battery 16.5Volts but your battery is only 10.95Volts. Your system has to boost the voltage up using a FET to step it up the rest of the way. But what would happen if the FET was bad? Then the batteries voltage would pass but its just enough to run your system (just slower when running on battery Vs the charger).
So your main logic board has a problem in the power section.
Are you up to replacing a few SMT components or do you just want to replace the logic board and be done with it?