Create fieldable bundle types for an entity with Drupal Console
I eventually got it by comparing how NodeType/Node is defined:

http://cgit.drupalcode.org/user_alert/tree/src/Entity?h=8.x-1.x

Now I can have unlimited entity types and add fields to them for different content configurations of the entity.

I started out by generating a config and content entity with the Console, but found that in order to get a more intuitive user interface, I had to modify the routing.yml and some of the paths as well.

Just a note for those looking for the same functionality.