According to Rav, Pinchas saw what was happening, remembered the halacha, and reminded Moshe of this halacha (in lashon hakodesh). Moshe then responded (in Aramaic) "Karyana d'igarta ihu lehevei parvanka" - the one who reads the letter should be the agent to carry out its contents. If Rav is quoting the exact conversation between Pinchas and Moshe, then why would Moshe respond in Aramaic (as opposed to lashon hakodesh)? If Rav is just paraphrasing the conversation and choosing a local colloquialism for Moshe's response, then what are the Bavel circumstances that led to that phrase coming about?
Chazal cite this expression on two other occasions in Shas (in Sanhedrin 96a, and in Bava Metzi'a 83b), the former with reference to Nevuchadnetzar, when he was the scribe of the then King of Bavel, the latter, with reference to an incident that occurred in the Gemara. In both cases, the entire conversation took place in Arama'ic, lending credence to your second suggestion, that 'Rav is just paraphrasing the conversation and choosing a local colloquialism for Moshe's response'.
Have a good Shabbos.