A bit of an update: we took Bilbo to the vet. He has an abscess (drained now) and is a bit dehydrated. The vet gave him some subcutaneous fluid, and we are having a blood test done.

Best guess: he got in a fight with another cat (most likely a cat given the almost invisible wound), hid somewhere, and developed an abscess. I doubt he was very far away during his “adventure”, but that’s just a guess.