Monthly Paris pass, living in Airbnb

Hi Bruce, you can look into ordering your personalized Navigo at Navigo.fr.  You don't need a "proof of residence" but you need to be able to receive mail & make online French debit card / French bank account payments all with French residential addresses.  If you're getting a French bank account setup, you should be able to order & receive a personalized Navigo card by mail (can take up to 18 days for delivery, so order with that in mind.)
 
I've read on Navigo.fr that monthly purchasers with a personalized Navigo card can also reload their card at home on a personal computer by buying (7€) & using a USB dongle attached Navigo card reader that can update the card's stored information, when connected to the Navigo.fr website.  Haven't done this myself as I only have experience with the Navigo Annual pass which is directly debited from a French bank account on a monthly basis.  If you're not good at remembering to buy next month's Navigo pass early in person (i.e. not waiting till last 2-3 days of month when renewal lines to ticket machines / offices are huge.), then this may be an interesting option.  But other than that, there isn't a notable monetary advantage to personalized Navigo cards vs. Navigo Decouverte.  Only the Navigo Annual passes get a discount on a yearly basis (12 months coverage for 11 months cost).