LIMIT_EXCEEDED (429) error when inviting user to a room on a (deep) hierarchical space structure.

This issue has been migrated from [#15882](https://github.com/matrix-org/synapse/issues/15882).

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### Description

When creating a space hierarchy of several levels and then inviting a user into a room in the deep hierarchy a LIMIT_EXCEEDED / Too many requests (429) arises. This is on single room / single invite not in multi invite on multiple rooms.

### Steps to reproduce

1. Create two users,
1. Then create a space hierarchy with first user down to 5 levels.
1. Then create a room into the 5th level of spaces.
1. Then try invite 2nd user into the room.
1. You will most likely get 429 error.

Using standard `rc_limits` i.e. fresh Synapse setup.

### Homeserver

Another home server running on UpCloud VM

### Synapse Version

v1.87

### Installation Method

Debian packages from packages.matrix.org

### Database

PostgreSQL

### Workers

Single process

### Platform

Distributor ID: Debian
Description: Debian GNU/Linux 10 (buster)
Release: 10
Codename: buster

Server running on VM in UpCloud

### Configuration

Fresh Synapse install, standard setups

### Relevant log output

```shell
Homeserver.log

2023-07-05 07:30:41,571 - synapse.http.server - 117 - INFO - POST-1016 - SynapseError: 429 - Too Many Requests
2023-07-05 07:30:41,572 - synapse.access.http.8008 - 482 - INFO - POST-1016 - 94.31.96.232
```

### Anything else that would be useful to know?

-

Reproduce the five-level space hierarchy and invite through the /_matrix/client/r0/rooms/{room_id}/invite endpoint, using the provided homeserver log as the starting evidence. Trace why a single invite reaches the standard rate limit and verify that the invite succeeds without a 429 under the reported setup.

Written by the indexing model from the issue text.