Customizing the display of properties on a rendered entity
We're trying to put an Entity Reference of this custom entity on a node and render only a few fields - with a form of preprocessing or something along those lines so that we can control the ultimate output.


Sounds like hook_entity_view is what you want. Use "teaser" or create a new view mode, and attach the reference using the given view mode, then implement your logic to render the properties as you like.

Alternatively / additionally, you could look at Display Suite and DS Extras which will give you more flexibility to expose properties via DS custom fields or in field templates.