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.

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".

This particular use of "to believe" and "to be" is recorded in Macmillan's dictionary

• transitive - to have an opinion about what is true or what might happen, although there is no proof
  believe someone (to be) something:
  His friends wouldn’t believe him to be capable of murder.

The infitival clause "to be …" has then the function of object. According to more recherché grammatical characterizations "The greater part of what my neighbors call good " is the primary object or recipient and "to be bad" is the secondary object or theme Wikepedia.

Therefore "to be bad " is a unit and that shows that "to be" is not an auxiliary in this particular use but a linking verb linking the object The greater part of what my neighbors call good" to "bad".