Is there a way of configuring the Issues component of Bitbucket to more closely parallel the difference between Severity and Priority as given in http://istqbexamcertification.com/what-are-the-key-things-in-defects-in-software-testing/ or is this something I will only find in JIRA?
No, the Bitbucklet team decided to keep BB tracker simple.
https://bitbucket.org/site/master/issue/5487/custom-fields-for-issues-bb-6719
Actually even JIRA by default only provdes a Priority field. A severity field would need to be manually created.
https://confluence.atlassian.com/pages/viewpage.action?pageId=192840
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