Need help distinguishing auxiliary 'be' vs main verb phrase 'be'

"The greater part of what my neighbors call good I believe in my soul to be bad."

Is the 'be' in the sentence above an auxiliary 'be' or is it serving a main phrase verb?

From my textbook, the auxiliary 'be' occurs before the present participle of the main verb, but 'bad' is not a verb so I am leaning towards it being the main phrase verb, but the phrase 'to be bad' seems to be adjectival as it is describing things his neighbors call 'good'.

Edit: My new position is that this is just that 'to be' is an infinitive verb and I am over thinking it.

The greater part of what my neighbors call good I believe in my soul to be bad.

I would advise you to avoid using the term 'main verb', since it's misleading.

"Be" is always an auxiliary verb, even when it's the only verb in the sentence.

Auxiliaries are verbs with the NICE properties (The acronym NICE means Negation, Inversion, Code, Emphasis). "Be" has those 'special' properties and hence is an auxiliary verb irrespective of its function in the clause.

Though "be" may indeed head copular clauses, it doesn't mean that it's a lexical verb elsewhere, it isn't; it's always an auxiliary.

Edit: you asked about "bad". Yes, it's an adjective in the infinitival clause "to be bad", where it serves as complement of "be". We say that "bad" is predicative because it refers to a predicand, in this case "the greater part of what my neighbours said".