Race condition when sending requests to join and set `displayname` at the same time

This issue has been migrated from [#10476](https://github.com/matrix-org/synapse/issues/10476).

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### Description

There is a race condition in the join and set displayname API requests to Synapse, https://github.com/vector-im/element-web/issues/18222

Comments from @tulir: https://matrix.to/#/!FDTDciUzgMWGJcYnwn:maunium.net/$n3crWcHOeSuprCGVp3tXMYHhNg8FZtcAQlDOxXxA5BI?via=maunium.net&via=matrix.org&via=tchncs.de


Let me know if any logs can help and I'll try fishing them out

### Steps to reproduce

- not sure, perhaps with a 'big' room (60 people) bridged from Telegram

### Version information

- **Homeserver**: matrix.chagai.website

If not matrix.org:

- **Version**:
"server_version":"1.38.1","python_version":"3.8.11"
- **Install method**:
docker/ansible

- **Platform**:
docker/ansible on centos7