3DDave…
I think you are far too pessimistic about the cost of suitable lenses for this purpose. Macro lenses are very highly corrected optically to produce a flat field and they exhibit very little to no pincushion or barrel distortion (the many distortion charts that I have seen in online review tests for macro lenses from several manufacturers bear this out). Modern macro lenses exhibit this behavior from closest focus to infinity, so they are suitable for more than just macro/close-up photography. Most macro lenses are also very affordable.
For example, I have three Olympus/OM System macro lenses for my Micro Four-Thirds kit: 30-mm f/3.5 (currently on sale at Amazon for $200), 60-mm f/2.8 ($400 sale price....I recently bought one for my oldest granddaughter's upcoming 16th birthday), and 90-mm f/3.5 ($1,300 sale price). The "full-frame" equivalent focal lengths for the same field of view are 60-mm, 120-mm, and 180-mm, respectively. If I was to do the OP's shoot, I would use my 30-mm because it has the widest field of view, which makes it easier to cover a large object like this gasket from a reasonable distance.
In addition, the mirrorless interchangeable lens cameras that I am familiar with all perform geometric distortion corrections in camera for their brand of lenses, at least on JPEG files. For example, a few months ago I bought an Olympus 12-mm f/2.0 lens for doing star trails and wide-field tracked exposures of the night sky. I knew that this lens had quite a bit of barrel distortion, but I wasn't worried about it, especially for my main usage. The barrel distortion clearly shows up the RAW files, but the JPEG files are fully corrected to virtually eliminate distortion.
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