I totally agree with HOS. I have flown two times to Cordoba via Santiago, both times on LAN CHILE. That is one fine airline, IMO, with the most accommodating stewards (no stewardesses on any of the four flights) I have ever seen.
I have had zero problems carrying my own gun(s). It is all very straightforward and easy. I hunted with Luis Sier both times and he provides a lady that meets you at the Cordoba airport and walks you through every step. She is a hoot, and the authorities know she knows her business.
This is just me now, but I would not enjoy the trip nearly as much if I could not use my own personal gun(s). Rented guns, to me, are for the very small chance that mine break and cannot be used. Hunting, and hunting trips, are a personal thing for me, and my personal guns add greatly to my enjoyment. My guns are an extension of me, and who I am. Rented guns are tools, period. Again, I have seen nothing in two trips there and back that would ever make me even consider not taking my own guns.
Hope you have a great trip. Drink it in deeply, it is an unbelievable sight when the sky is full of doves, constantly, for three plus hours of shooting as fast as you can reload.
All my best, SRH