Thanks to all who replied. It seems most likely that this is the
retaining ring that holds the door seal against the outer metal skin of
the machine. The ring goes into a groove there of just the right size,
and its little angled teeth lock it nicely at about 7pm.

Though how it got in with the laundry is a mystery, for the (closed)
porthole should have kept it firmly in place and excluded it from the
drum...

I have run the machine, watching it carefully, and it did not leak. Nor
the the spinning noise recur, so the bra wire must have got lodged
somewhere harmless (for now?!).

This does make me suspicious that it might have come from somewhere
else. But I don't see where, because the door seal is well connected to
the outer drum and the door. The only other suitable-looking place I
can see is the inner edge of the door seal, where it mates with the
drum. Is a locking/sealing ring needed there?

Skylark