The subjunctive If I were does not apply in this sentence; to use it would be a case of hyper-correction.
The "If I was" part is not actually from the speaker's own hypothetical point of view about himself. What that clause is doing is referring the point of view of the parents or guardians, which existed in the past.
Prior to the speaker's birth, those people said to each other something like, "If it is a girl, we will call her Kate". This is not a subjunctive mood, but a straight conditional: it refers to an unknown future, not to an imaginary alternative present or past.
The speaker is relaying this conversation between the parents or guardians, using reported speech rather than as a direct quote. In doing so, the point of view shifts to the first person, and so "it" becomes "I", and "is a girl" becomes "was a girl".
"If I were a girl" is something else: it is a statement referring to an imaginary present, from the speaker's own point of view. It is not applicable in the sentence, because the sentence is about a naming decision which took place years ago: that naming decision is not conditional on an imaginary present; it is conditional on different events having taken place in the distant past.
"If I were a girl" can only be properly used as a conditional for present and future statements such as, "If I were a girl, I'd be wearing a summer dress right now". "If I were a girl, I was going to be called Kate" is nonsense, because "was going to be called Kate" is situation which really existed in the past, whereas "If I were a girl" is an imaginary present: thus, an actual past is made conditional on an unrealized present. On the other hand, "If I were a girl, I would have been called Kate" is possible. This hinges together an imaginary future and imaginary past, which is possible ("In an imaginary world in which I am a girl, my imaginary past is such that I was called Kate"). This sentence now has a different point of view, though; it is no longer relaying the circumstances of the naming decision made by the parents, but simply remarking on an imaginary past and future.
The sentence "I was going to be called kate if I was a girl" is correct; its only flaw is that it is clumsy. It is not making effective use of the nuances which are possible with English tenses, as well as alternative vocabulary, for maximum clarity. A rewrite such as "I was going to be called Kate if I had turned out to be a girl" is no more correct, only more polished.