The problem is isolated to the display subsystem, but do you see a faint image, or is the screen completely dark? If you see a faint image, you most likely need to replace the inverter cable (which goes bad infinitely more often than the inverter itself). If that doesn't do it, then the screen itself is the issue. You've proven the GPU works by successfully using an external display.
I don't agree with the "backlight is definitely powered from the video processor" statement...the GPU doesn't have a lot to do with the backlight...it's the screen electronics, inverter, and inverter cable that light the screen, and when any of those are bad, you'll often see a faint image with no backlight. In this situation, the GPU is working fine and doing it's job, providing the image, but the backlight is not its problem.
There are cases where a logic board is bad and prevents a backlight from working (even when the screen, inverter, and inverter cable are good), but I don't believe those cases are specifically related to the GPU...I think this situation has more to do with the logic board failing to provide power to the inverter, and, again, I don't believe the GPU has any part of that.