Update the Bitbucket tests to use the new hosting service test helpers.
Update the Bitbucket tests to use the new hosting service test helpers.

This updates the Bitbucket test suite to use the new helpers for hosting
service unit tests, which provide better, more consistent checks with
simpler code. The results of all HTTP requests are checked, ensuring
we're calling what we expect and that behavior doesn't change in the
future.

Testing Done:
Unit tests pass.

Reviewed at https://reviews.reviewboard.org/r/9778/

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