After a substantial amount of banging my head against the wall I had to take some time off from this project to recover. Ultimately the correction process using the camera didn't work just because the camera (even with random plastic filters) doesn't respond at all accurately to the wavelengths of light that alt-process prints do.
I decided to have another crack at it this weekend and have made some good progress now. I decided that the only way to accurately record the wavelengths the print actually cares about was to make prints. So I made a print of a flat grey rectangle, which very clearly showed variation in density relative to the variation in the light source. I then took a photo of this print with a copy stand and used it as the flat-field correction image (after converting to B&W, flipping, inverting and applying my cyanotype calibration curve to it). I made another print of the same flat-grey rectangle but with the correction applied and the vast majority of the non-uniformity is gone.
Uncorrected on the left, corrected on the right.
Obviously not perfect yet, there's a lot of negative-vignetting which is probably a result of how I digitised the image (digital camera on a copy stand with two LED light panels either side) and some weirdly dark areas top and bottom. I'm quite confident I can fix these issues however, I'll just repeat the process of digitising the 'corrected' print and add that to the first correction image so that the result is a combination of both.
I knew that this was probably the best way forward early on but I was really trying to avoid it as it's quite slow, having to wait 24 hours for the print to dry and darken each time I want to iterate, but glad that it's working out.