I photograph paintings for reproduction using carefully set up and balanced lights. Lens profiles do a pretty good job of compensating for lens fall off, but I want to compensate for slight variations in lighting over the artwork area and also an inherent sensor falloff that seems to exist in my camera. Robin Meyer's Equalight accomplishes this but it's a clunky and time consuming workflow. Capture One Pro 8's LCC - Lens Cast Correction - is a more elegant and efficient solution. Does Lightroom not have the ability to analyze a white frame capture and apply the adjustments to other image files?
We could paint the area on the white frame which is affected by the lightfall, and then you can make adjustment to the area.
Check out this tutorial for the same Adjustment Brush: the basics | Adobe Photoshop Lightroom CC tutorials