What is the grammatical function of “a Catholic” in “She was raised a Catholic”?

1. In his mother was raised a Catholic, the phrase a Catholic is a subject complement. A subject complement serves to assign a property to the subject (or sometimes to describe identity, but not here). The verbal phrase be raised can take a simple subject complement, just like other passive forms of verbs that normally take an object and an object complement:

I considered her a Catholic → she was considered a Catholic (by me).

The odd thing about be raised is that the active form does not always take an object and an object complement, especially not in informal speech; and so the passive form with a subject complement is also less common.

I raised her a Catholic → she was raised a Catholic (by me).

2. Using as a Catholic is more common in modern and informal language. There is no difference in meaning. It functions as an adverbial phrase of manner or circumstance. You could say as functions as a preposition here; but, more traditionally, I believe as is interpreted as a subordinating conjunction with an elliptical clause:

I raised her as a Catholic. → She was raised as a Catholic [is raised].

3. In to be a Catholic, the infinitive to be functions as an adverbial complement of purpose to the verbal phrase was raised. There is a subtle difference in meaning, in that it focuses on the result of the raising rather than its manner, but this is often of little consequence. It is a common construction, similar to other verbs that can take to infinitives of purpose:

I constructed this building to be a church. → This building was constructed to be a church.

I instructed the teacher to be strict. → The teacher was instructed to be strict.

I raised her to be a Catholic. → She was raised to be a Catholic.

The correct interpretation of "Raised an X" is "Raised as an X".

If one is "raised to be an X" then that would mean that they are being trained to eventually graduate, as it were, to become an X. E.g. "raised to be a member of the King's guard".

Whereas "raised as an X" would imply indoctrination from an early age or birth, as happens with religions or nationalities.

So one might say "I was raised an Arsenal supporter", but "I was raised to be a footballer".

"A Catholic" is part of the predicate of the sentence.

(In isolation) Subect: His mother. Predicate: was raised a Catholic.

It simply means 'His mother was raised as a Catholic'. By grammatical convention the 'as' can be omitted.