Put a tinted glass on the screen
No, the inverters are usually designed to provide the appropriate voltage for the lamp, if they are feed on lower voltage, they try to cope until being unable to do.
They will increase the width of the impulse to hold the voltage they are designed to.
If you are not sure about your inverter fits the Lilli, unplug it VERY quickly, because it can destroy the CCFL!!!! And it is not easy nor cheap to replace...
It won't destroy the LCD (Unless it produces much more heat).
Perhaps your inverter has a third input line (besides 12V and GND) to regulate the brightness.