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.
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2014-09-10T18:37:14.153
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2021-11-28T07:03:40.350
2020-06-17T08:18:27.500
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question
grammar
How would a name be used as a non-restrictive appositive?
Just like you said 「[富士]{ふじ}の[山]{やま}」, you can say 「[父]{ちち}の(his name)」. That
is completely natural and grammatical. 99% of the time, we would not add a
pronoun and say 「私の父の(his name)」.

「Bという私の父」 does sound more than strange. You have a good ear.

「Bという父」, however, would be possible in certain contexts. You could use it if
you wanted to emphasize your father's name, for instance, in a composition or
speech in which you talk about your father. You could say something like:

「(father's name)という父を[持]{も}って、[本当]{ほんとう}に[幸]{しあわ}せです。」

Obviously, though, that is not something you would say very often.
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2014-09-11T01:23:56.940
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2014-09-11T03:11:09.460
2014-09-11T03:11:09.460
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answer
“私の父、B” is common in modern Japanese.

“Bという私の父” sounds as if you have several fathers and one is named B.
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2014-10-24T14:02:48.587
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2014-10-24T14:02:48.587
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answer