Profile change just before leaving a room may cause a rejoin

This issue has been migrated from [#12926](https://github.com/matrix-org/synapse/issues/12926).

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This is likely the "mirror" manifestation of the problem described in #10066. Feel free to close and track it there instead.

### Steps to reproduce

- call `/profile/{userId}/displayname`
- then immediately `/leave`

Assuming permissive join_rules, the user leaves the room first but then immediately rejoins, as `/profile` call catches up going across all the user's rooms. Unreliably (as it's a race) reproduced on current matrix.org.