I guessed which chip it was (with a little bit of help from leaked schematics for similar laptop models), bought the first programmer that I saw online that appeared to be compatible (anything with a CH341A in it, apparently) and ran `flashrom --programmer ch341a_spi -r bios.bin`.
Had there been any more than one possible type of chip and more than three of these similar looking chips on the [accessible part of the] motherboard, I'd probably still be sitting here trying to figure out what to do.
In any case, I'll try and add it to the blog post once I figure out how to do footnotes. :^)