Backbone offers a REST API in a Jira Server/Data Center environment. Our users should also be able to use the REST API in a cloud environment.
Backbone's REST API is also available in a Cloud environment under the base url https://backbone-issue-sync.addons.k15t.com/api/1/*. Not all endpoints are available, but they're really similar to Server/Data Center. But there's no easy way to allow authentication for a user yet.
As a manual workaround, you can inspect the network calls made by Backbone's UI via the developer tools in your Browser and get the X-Authorization header with the JWT token content. If you set this header in your requests, it should work until the JWT token times out.