Thanks all!
The stains may not be glue -- they could be some kind of grease, perhaps? -- but unfortunately they will be visible -- maybe this photo shows them better:
Anyway, good call about the possibility of white vinegar bleaching the wood, Simon! I've swabbed some white vinegar into the bridge rout, and some rubbing alcohol (70% isopropyl alcohol, no idea what the remainder is but hopefully water) into the neck rout. Will leave that overnight to make sure there's no damage, and if all is well then let's hope one or the other will lift the stains.