I worked it out.
The recent MW3 update changed the game from 32-bit to 64-bit, and Plutonium IW5 currently doesn’t work with the new 64-bit version.
Luckily, on my system I found a copy of the old 32-bit files here:
%LOCALAPPDATA%\Plutonium\game-files\iw5-legacy
I pointed Plutonium at that folder instead, and it started working again.
If you don’t have that folder, Plutonium staff have also posted instructions for downloading the older 32-bit IW5 files through the Steam console.
Hope this helps anyone else who runs into it.