Fix error logging with `GitlabSettings::Options`

https://gitlab.com/gitlab-org/gitlab/-/merge_requests/122888 added
logging that did not work properly. Clean up the exception handling to
use `Gitlab::ErrorTracking.track_and_raise_for_dev_exception` and fix
the test.

Relates to https://gitlab.com/gitlab-org/gitlab/-/issues/415968

Changelog: fixed