Expanding my comments in an answer:
The fact that the language of ham operators is to some extent standardized does not prevent us from analyzing it, especially since it is clearly rooted in natural language. Similarly, linguists use European Parliament proceedings for corpus research, and that language can be formulaic as well.
What I would do is look for parallels in more natural language. The use of CQ in CQ, CQ, CQ, this is ... is similar to a greeting, like Hello in Hello, is anybody there? Since hello is analyzed as an interjection, I would analyze CQ in the same way.
In calling CQ it's originally a quotation, but you could say it has been incorporated into the verb (i.e., there is a verb call CQ). Similar things happen with play soccer (though here it's a proper noun that is incorporated into the verb, rather than a quotation).
You mentioned:
Perhaps it is effectively a plural "you"? At the start, is it also a noun or pronoun, used for direct address?
This is an option, but it is not attractive. CQ only appears at the beginning of a call or in calling CQ (where it is clearly a quotation). So it would be limited to the vocative (as you note) and, perhaps in calling CQ, object position. But such a pronoun could normally be used after a greeting (hey you), which is not possible with CQ (*hey CQ). Furthermore, CQ is a bit odd in that it has no referent. When calling CQ, you often don't know if anybody is listening, so it is not clear what a 2nd person plural pronoun would refer to. In the comparable situation that you enter a shop and there is no one at the counter, you would call out Hello? rather than You?, so again, CQ feels more like an interjection to me.