They won't ask for your passport to prove your residency -- they will ask for your DNI or Libreta -- or do you only have temporary residency? (in which case I'm not sure if that is a paper in your passport or what) -- you will need proof of your residency to be able to fly on a resident-priced fare, otherwise they may make you pay the difference at the check-in counter.
A driver's license may not be accepted, do you have a residency card yet? I use the card form of my DNI for all internal flights. Or do you at least have an Argentine DL? Make a call to Aerolineas before you send off your passport and see.
What you may end up having to do is get a notarized copy of your passport with whatever evidence of your residency status it has in it and use that for the time being.
