"Have at it" : what's the object of the transitive verb "have" here?
Solution 1:
I believe that your analysis is mistaken: to have in this expression is intransitive.
Of course the verb have is normally transitive, which is why you assume that it must be transitive here. But I would argue that the to have at is a phrasal verb, whose object is syntactically the object of the preposition at. This is similar to a multitude of other phrasal verbs, such as to look at.
So to have in this idiom is intransitive, and the notional object is expressed as the object of the prepositional complement at.