Had a look at your correction frame with “Digital Color Meter”, an app that comes with macOS:
The frame has a falloff that ± corresponds to the bright edges in your first sample screen.
The falloff seems to not be important enough for Lightroom to recognise it when you tried flat field correction…which is a pity because one would assume that FFC can also correct minute differences.
According to what I found in respect of FFC, I assume that you took the reference frame with the same settings as your negative(s). Changing something (distance, f-stop and whatnot) might make Lightroom reject the reference frame based on metadata…but that is my guess only, but it might be worth exploring.
As for the illumination of the negative: Intensity drops inversely proportional to the distance between the light source and the object. This also takes a toll with diffused light, specially when two diffusers are used. This means that the first diffuser (as seen from the light source) must be bigger than the second diffuser, which should be bigger than the negative (unless the distance to it is zero).
Diffusing the light helps to even illumination of the negative, but the diffusers need to be big enough in relation to their distance and size of the illuminated object or negative.
Here’s how light levels look on a model panel with 15 LEDs in a row:
(made with “Grapher”, a utility that comes with macOS)
Note how neighbouring lights add to overall brightness. While the central light outputs the standard “1” (in arbitrary units) like all others, the overall central brightness is almost 3 times higher. Better evenness can be attained by moving the diffuser further away from the lights, see real world example:
Left: Diffuser way too close to the LEDs. Right: Better, but still too close.