Monsta's tip did the trick seemingly; my freshly installed LMDE Cinnamon 201303 spin matched the output in his post perfectly, so I ran the command:
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sudo apt-get install debian-system-adjustments
And the update went forward with one troubling symptom:
In the middle of applying the various updates, Update Manager just quit, taking the cinnamon desktop with it, leaving only the background wallpaper.
I switched to tty1 and was able to re-run startx, but from there I did a system shutdown, powercycle and restart (the machine was still on the 3.2 kernel, not the 3.10). The machine came back up, but when I opened Update Manager, I got an error that said I had to run:
which completed the configurations of the modules. This process is running even as I post this, and I am confident that it will complete properly, though I will do a reboot afterwards just to make sure.
I am trying to figure out why my LMDE XFCE box did not have this problem when I applied UP7 (aside from the fact that it isn't running Cinnamon), and I can only surmise that the debian-system-adjustments package got updates (there have been a dribble of updates via Update Manager in the last month or so).
On the one hand, I want to thank everyone for their help, but the fact is that the preparation steps Monsta listed really should have been included in the UP7 notes (which can be reviewed prior to installation) is a very egregious oversight on the part of the Mint team.