Flat Field correction

I applaud your inventiveness but AFAIK flatfield image function was never intended for this job. It is AFAICT meant to address sensor fall-off, as distinct from lens related vignetting, hence belongs in the capture realm not in the scene realm.

Would you consider instead opening both images, together, into Photoshop "as layers" - and using a layer blending mode there, to apply a reverse tonality of the white-card image over the top of the painting? (may need to blur the white-card if it shows any texture)

I don't know if this is about a repeatable and controllable studio setup, or a one-off situation. In the former case there are of course some established methods for getting sufficiently even physical illumination. A custom lens profile could only help IIRC with radial unevenness, so its vignetting correction can probably not be misappropriated for the task even at the cost of a lot of work.

Personally I would probably just concentrate on studio lighting in the hope of reducing the need for any correction. External editing is incredibly powerful but comes at some workflow cost, IMO most significantly the need for careful thought about which LrC operations should happen before-(onto the camera Raw), and which should happen after (onto the resulting TIFF/PSD), and what that means for the external editing settings to make those technically appropriate.

I have since found out that the reason it doesn't work, is that Lightroom refuses to do it if the flatfield image is not completely "flat", i.e. in my case if there are edges around the whitecard showing.

That is impossible to prevent out of camera, and cropping the raw image doesn't solve this.

I have found a workaround, but it is quite cumbersome and I'm not sure if it will preserve the quality, curious if anyone has a better idea.

The workaround is to crop the raw files in Lightroom (i.e. crop the painting and apply settings to the flatfield image), export as tif, and then reimport those. Now the crop is baked in, and it does work.