Bitbucket requests had no concurrency cap and no shared pause, so after the
first 429 every queued branch lookup still went out and got its own 429.
They now share four request slots and the host-wide pause GitHub already
uses. Background reads fail fast while paused; writes and interactive reads
still go through.

Closes #23885

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>