It's a modal (judgement-indicating) modifier and is best seen operating in the expanded paraphrase (OP justifies a male player):
"I am with the man who is the best basketball player in the nation. (The statement [this man is the best basketball player in the nation] is open to opinion.)"
So it's qualifying the whole statement [this man is the ...
One way of figuring it out is to "divide and conquer", that is, to take a smaller part of the sentence (a constituent part) that still contains the word in question, because this will rule out some of the words that definitely aren't modified by it.
For example, with your sentence, i could do it like this:
I am with [arguably the best basketball ...
Different verbs require different means of adding complements to them. They may also differ based on whether the complement is a noun phrase or a verb phrase and may have more than one possibility, depending on the precise meaning. The differences are loosely connected to the meanings of the verbs, but are not really predictable.
You can say:
I want your ...
Yes, copular VPs can be modified by PPs, cf. I was happy when I met
her / I was at that time extremely happy / I was when I met her much
happier than I had been for several years (note that I take "when" to
be a preposition). The natural interpretation of the PP in your
example has it as an adjunct in clause structure.
Answer provided by BillJ.