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GH-36523: [C++] Fix TSan-detected lock ordering issues in S3 (#36536)

It is counter-intuitive, but lock ordering issues can happen even with a shared mutex locked in shared mode.
The reason is that locking again in shared mode can block while there are threads waiting to take the lock in exclusive mode. For this reason, we must avoid to keep the S3ClientLock taken before is it taken again.

* Closes: #36523

Authored-by: Antoine Pitrou <antoine@python.org>
Signed-off-by: Antoine Pitrou <antoine@python.org>