The problem is going to find sufficient footage to make any identification, I only know of a few pictures and they are very grainy. These cars were hand-built to order, but most of the details would be either in the engine bay or interior and color, all but probably the color (perhaps a white) is missing from the picture. If the color/paint job days changed, often the curators will detail the original color in listings which can be gotten from hidden parts of panels where the paint job missed a spot. One thing I can note on the Rutherford pictures is the badge on the ‘spare’ wheels, which indicates the V16 emblem and if not replaced/restored over time, it could narrow down the search.
It is likely a V16, those were the most popular models if you could afford them, I would think Rutherford would rather have a V16 than 2 V8s like the ‘plebs’.
Then again, thousands were built and only a few hundred survive. Given the borg, I wouldn’t be surprised they trashed the thing to bury the evidence after his passing.
A model closely resembling both color, body features etc was sold in 2017: https://www.classic.com/veh/1930-cadillac-v-16-imperial-limousine-nlm7aMW/