There is a "manifest" REST API endpoint for this that is undocumented and seemingly broken. I found it via the apigee browser:
(The apigee API browser suggests a trailing slash, which didn't work).
Anyway, this will only list the files at the root of the repository. I don't know if it's possible to make it behave recursively.
Is there another way to list all files in a repo (for a given revision)? We'd very much like to support Bitbucket in https://cucumber.pro, but we need this functionality to do so.