Thanks for the link! Turns out I had already been there and tried all these. But when searching for a firmware update (still none available), I noticed there was a driver a month newer than the one I had. I installed it on a couple machines, set the printer's Sleep timeout to 5 minutes, and waited. When it went to sleep, I printed to it. It woke up, and printed fine.
And it hasn't gone back into Sleep mode since last Thursday (including while we were closed over the weekend), despite the timeout still being set at 5 minutes. Apparently something in the new driver disabled Sleep mode altogether. Not quite the same as waking up and printing like it should, but probably the next best thing.
Thanks again!