Question

Trumpets and trombones are part of our band.
In some ways this feels like part is a direct object, but it also seems like this could be an intransitive sentence.

Answer 1

Let's start with the verb, "are." It is always either a linking or a helping verb -- so it is intransitive. It takes a predicate noun (or nominative).

The predicate nominative is "part." It means the same as the subject, "trumpets and trombones."

Answer 2

Consider the examples:

I hit your nose.

The word nose is obviously the direct object of the transitive verb, "hit."

Your nose is red.

The intransitive verb, "is," links the subject and the adjective that describes it.