Chosen Solution
I have an iPad Mini 4 whose broken screen I replaced but backlight won’t light. When I connect the backlight and the digitizer to the board and then plug the battery in I get a small flicker of light and then it goes dark. Is this an issue with the backlight filter or the backlight connector?
Have you tried to re-connect the original, broken screen? If you have the same symptoms, then you probably have an issue with the backlight circuit. Based on what you say, it does appear to be a flaky backlight filter. I would inspect it to insure it is properly soldered on the board and check it for continuity.