For our trip to Switzerland next week, Stella remembered yesterday that maybe we should look into what kind of travel documents we need—since her passport expired just over a year ago.
Turned out that German citizens can enter using their identity card (phew!) but that children need an identity document as well if they're not entered in a parent's passport.
I phoned up the local council office and they said that Amy would need a children's passport, but that that'd only cost €13 and they could issue one while we wait, which was good. So we got some passport photos, went over there, filled out a form saying that both of us parents were fine with the child's receiving a passport (presumably that form was there to prevent child abductions where one of the parents takes off into another country with the kid), the lady printed the information onto special paper and stuck it into a blank children's passport, which she then handed over to us. The thing is valid until Amy is 10, though if she changes enough not to be recognisable any more, we ought to get a new one.