Add Field as Entity in Group Module
I am not sure how you would approach this by adding the fields as entities via the GroupContentEnabler (that'd mean one custom plugin per field? Given a large number of fields that might not be the best solution). However, for an example how to implement your own custom group-content-type plugin see this answer.

Another possible approach that might be worth checking is to implement hook_entity_field_access() to define access checks for single fields programmatically (that's how I've solved a similar problem). See the group module's gnode.module file for an example how to check permissions/roles for users and add cache-tags to access results.