These other apposition-with-name variants are readily apparent to me:
Name as appositive genitive
富士の山 = the mountain of Fuji
Name modified by non-restrictive appositive
私の友人のAさん = A-san, my friend, / A-san, a friend of mine,
私の父であるB = B, my father,
(Note: I know that the above two are often translated as the below, but I'm interested in grammatical equivalence, if possible.)
Name as restrictive appositive
Aさんという私の友人 = My friend A-san
What about a name used as a non-restrictive appositive?
? = My father, B,
? Bという私の父 doesn't sound right to me, but my ear could be deceiving me.