PAS, in the USAF, also means Personal Accounting Symbol, which is what's used as the accounting code for every person assigned to a specific unit on a specific AFB. Every unit on each base has its own PASCODE (with the first three letters signifying the base itself), and it's used to determine whom to bill for permanent change of station moves, temporary duty assignments, and other personnel-centered AF decisions that generate bills.

I have no idea how long PAS as its own code has existed. But if such a code existed in the Army Air Corps (or any military branch, for that matter) when commercial aviation was just beginning its boom, PAX may have stuck for "passenger" to avoid confusion with other acronyms, since both a Pax designation and a PASCODE can exist simultaneously on a set of orders - and then like a lot of military aviation speak, it translated over to the civvy side, as I'm sure most early commercial pilots were ex-military aviators who dragged familiar words and terms with them.

Just my two cents. I find etymology, even for the mundane, very interesting.