Networking with detailed user permissions

I’ve been tasked with making a website for a local animal welfare organization for the purpose of connecting rescue groups with willing foster homes in the community.

Visitors should be able to apply to be a foster home or rescue group, pending admin approval. Approved rescue groups should then be visible to the public, while approved foster homes should be visible only to approved foster groups– and admins of course.

I’ve got Buddypress and several “Membership” plugins activated but I can’t seem to find anything with networking user permissions that are detailed enough to accomplish this task.

Any suggestions?