Thanks for the reply. I was able to connect to the database by applying only CONNECT permissions. There are still some things that don't look right to me.
The 3 other databases, with CONNECT only permissions, showed relationship classes. However, the database in question was empty (even though there are multiple relationship classes just like the other databases). So I guess it isn't connecting to the database that is the problem rather viewing permissions once connected.
With CONNECT only permissions I next added Feature Dataset permissions through ArcCatalog. Three of the databases seemed to work correctly giving me access to the particular feature dataset (see screenshot):

The database in question allowed me access but something is wrong:

That should all be contained within a feature dataset (and why is it showing the historical tables and version views?).
Following your instructions I created a brand new login and got the same results as before.