How would I know it was a Buckingham elf, though? I don't think they walk out the front door.

I'll come up with a list of places the Palace elves might feasibly be sent to and keep them in mind. Once I'm on my next assignment I might actually be in a position to find out more.

I really don't know anything about how house elves work. But I suppose I could ask Heddy, Gibby, or Shilly. Those are the Malfoy elves that are still at the Manor.