There is no more elegant way to do this. A computed column cannot rely on an external function (even if you hide it behind a synonym, which seems to be the kind of "shortcut" function you're after) - the problem is that schema stability can't be broken by cross-database relationships - what happens when the database is offline, or single_user, or gets dropped? This is similar to the reason a foreign key can't reference a table in another database and a table can't belong to a schema in another database.
You already know that you can't use a 3-part name (in SQL Server 2014 I get a different error):
Msg 207, Level 16, State 1
Invalid column name 'other_database_name'.
If you try to mask your function with a synonym, you get:
Msg 2788, Level 16, State 2
Synonyms are invalid in a schemabound object or a constraint expression.
So, deploy your function to all of your databases, or pre-calculate the values and insert and store them manually instead of relying on the computed column.