Holy App Type, Batman! It works! 
The piece I missed was that an “INPUT” (field / property) of type “App Type” must first be defined before that type is available as an “OTUPUT” type (the context being an exposed state in the case below).
As for using it in a published plugin, I’ll have to give that some thought. I generally eschew detailed docs in favor of a friendly and approachable UI. In this case, however (and since the audience is a Bubble dev and not an app end user), I might have to reconsider. Despite it being a bit convoluted (and confusing at times) to set up, the benefit gained in the editor might make it worthwhile in some cases.
As @AliFarahat said, though, I hope Bubble makes it easier to do this in the future!
Thanks again, @lottemint.md, for working this out and posting it!