What feature do you want to see added?

RestAPIExtension to extend AuthData through rest api (Just like MySQLExtension).

E.g. send JSON with {"user: "xXSchrandXx"} to https://test.test/extendAuth and receive a JSON with extendAuth information (Like the password and salt).

Are there any alternatives?

Add MySQLExtension for any cms.
Don't use Authme with managed CMS.

Anything else?

Some managed CMS provider deny direct database access. With a plugin on the webpage to process the requests, this could be solved.

Linking the CMS account and Minecraft name can be solved with other plugins, too.