Mobile Device Updater stuck on downloading

Hi folks! I am dealing with some aging equipment here, and I suspect that my 9.7 inch iPad Pro is on the way out. For one thing, it shuts down when I leave it for a few moments, and I have to force restart. I have tried resetting settings, as well as a full restore, and they helped for a few days, and then back to square one.

I also have an aging late-2015 iMac, and a combo problem between the two: I recently (finally) updated to Catalina, and my iMac has that alert to update to allow communication with the iPad: https://support.apple.com/en-ca/HT208831. But it keeps getting stuck on “Downloading” with a full progress bar (I am talking for literal hours). The iPad is showing up in the Finder ok, so it seems that it can communicate with it. But (almost) every time I hook it up, I get this same alert, and the same stuck on downloading.

Obviously, both devices are WAY past support, so any suggestions would be welcome! Thanks so much!