I've run into this issue before.
You need to strictly override some of the views in there by clonning/forking the project and install it from your own location, because as you noted, postman is template-oriented because it's meant to only get the needed templates configured and a few settings. I mean, the backend is meant to work as is.
What you need to do is override stuff like:
Message
model's recipient field to be aManyToManyField
- customize the views based on your needs and be careful with
Message.replied_at
- make sure you allow a user to reply to their own messages (by default, it was not allowed when I ran into this, not sure now)
Depending on your needs, maybe you'll want to override something else, but this is a good start. If you need it facebook-like, you'll need to use some push libraries as Pusher or Juggernaut, maybe you're interested in them also.
Good luck! :)