[cmags]

There shouldn't be an issue with powering the backlight off a separate power source, but make sure it's a regulated 5V. You can pull the 12V from the Accessory circuit (ACC) in your car which switches off with the car (and during an engine crank). Easiset place to pull ACC is behind the radio, but there are many other things in the car that run on ACC.

Edit: Sorry, forgot you had this powered via signal + transistor (don't have a backlit LCD yet). But yea, that would be a software fix. The above still applies if you don't want to use the software option for brightness control though.