You haven't mentioned whether you have probed anything yet. I would work backwards probing from the LC display until I find out where the drop in display voltage/current is coming from. Could be the driver ICs, could be a voltage regulator feeding the IC, or something else. I didn't look at the datasheets on these ICs, but one of the ICs/regulators may have thermal or overcurrent protection built-in. If so, then once you isolate the device you can check for downstream from it for unexpected low impedances and start probing back to the LCD. Assuming excessive current draw, it could be coming from anything such as an electrolytic cap that is aging or something as simple as a barely bridging solder joint and/or excessive flux that after enough time has started to cause excessive current leakage.
I'm no expert with LC displays either, just my 2 cents.