“whose” vs “who/that + possessive”

I agree with everything that John Lawler said, but there is something I would like to add (so I will not repeat the things he's already made clear).

Whichever option you choose, the important factor that would determine whether him is compulsory or not licensed is the direct object of the verb call. The person these kids want to call 'daddy' is a man. So in the relative clause this direct object must be expressed by some kind of pronoun that would replace a man. In the version with that:

a man that his kids want to call 'daddy'

that his kids want to call 'daddy' is the relative clause and that has two functions:

1. it modifies man as head of the relative clause
2. it also serves as direct object of the verb call

Another pronoun which could replace it is whom:

We use whom in formal styles or in writing to refer to people when the person is the object of the verb (of the relative clause). We use that instead of who, whom or which in relative clauses to refer to people, animals and things. That can act as the subject or the object of the relative clause:

• It’s the same cooker that my mother has. (that refers to the same cooker and is the object of has in the relative clause). (Cambridge)

So your relative clause that his kids want to call 'daddy' already has a direct object: that. To add him is not grammatical because it would express the same object twice. You cannot say:

*This man's kids want to call this man him 'daddy'.

Even if him referred to a different person (to another man, not the father of the kids), *a man that his kids want to call him 'daddy' is still ungrammatical because in this case that is no longer licensed ( it would modify the subject which is already modified by his) and should be replaced together with his by the possessive whose.

As for the other option, if we replace that his with whose, we have:

a man whose kids want to call 'daddy'

The relative has now become whose kids want to call 'daddy' and whose also has two functions:

1. it heads the relative clause modifying a man (just like that in the first option)
2. it replaces the possessive determiner his, modifying kids which is the subject of the relative.

So the verb call is now left without an object, because as we showed, whose modifies the subject:

*a man whose kids want to call 'daddy'

But to call is transitive and does require an object, so if we want this NP to be correct, we must add it (him):

a man whose kids want to call him 'daddy'.

That is not to say that whose cannot also modify the object of a relative clause (it is just not the case in your example):

You’ll use the possessive case whose in those clauses that have their subject and their object already satisfied and don’t need an object of a preposition. Thus:

• The child whose homework the teacher graded first received an A. (whose homework is the object of graded). (Grammar.com)