All that said: problem solved, apparently? I replaced the version of SMW which was in the MW install (how did that even get there) with a link to the one I had explicitly downloaded earlier.
Things magically started working.
It looks like they're really just not supporting MW 1.36 properly, and I just upgraded to 1.37. SMW development seems to be kind of in a holding pattern right now.
I commented out 3 lines of code around where the error was happening, and that seems to have at least made the site viewable (I haven't tried editing yet).
If SMW would like to hire me to fix stuff, I'm available. :-P
On the internet, everyone knows you're a cat — and that's totally okay.