The Atlassian REST API Browser is not (marked as) compatible with Bitbucket Server 5.0.x.
Since the plugin is developed by Atlassian Labs, and not supported, how does one get it to support new Bitbucket Server releases? There is no link to the source for the community to contribute to its development.
The plugin is referenced in the Bitbucket Server REST APIs documentation, so it's very disappointing that it's not officially supported.
For anyone running into issues here, I'm on Bitbucket Server 6.2 and I had to go to the rest-api-browser source for version 3.2.3, atlas-package, then install the jar using the plugin manager. It works perfectly and should support Server up through 6.2.