Hello Matt -
(and welcome to the forum)
Hmm, that is quite interesting. When the backlight is on, does the image show up correctly (also in windows), or does it look something like what Illgrill have? If it looks fine, then you actually might be able to help Illgrill.
When the backlight is on, does it have a red/pinkish tint at the left and right sides of the LCD?
Does it have some sort of manufacturer date on it? If yes, how old is it and does it look used?
Theoretically the CCFLs for larger LCDs require a higher starting voltage to light up compared to smaller LCDs, but when the difference we are talking is between 14.1" and 15.0" LCDs, i would say it does not matter at all.
When the backlight shuts off it is actually due to a protection feature by the inverter board. When CCFLs get used, the require more and more power to turn on (when more and more mercury is depleted in the phosphorus layer). When this happens the current consumptions explode, and the CCFL will eventually heat up until they melt near their ends. A current limited power supply is therefore a MUST. This also applies to normal cathode tubes.
No-one can really say if Quanta used some sort of weird CCFL for this, with unusual requirements, so it's really hard to say. But the odds are that it's a bad or old CCFL
If the inverter doesn't work with your XGA either, try replacing it with a new inverter (for 14.1")
It would be nice with a data sheet, but Quanta is long gone - it was acquired in 2006 by AU Optronics.