Have done same itinerary flying into Eagle for hut trips. You can fit a lot of clothing in a ski bag and it protects the skis. Ditto, buying all food and other perishiables there. Also, ditto the backpack with some stuff as carry on and boots in hand (a PITA) but saves a lot of room in luggage. After that, there's not just that much stuff...a lightweight sleeping bag, and other stuff should fit in one bag, easily let alone two...
Have fun, haven't skiied 10th mtn in a few years, what an awesome hut system... also borrow some gear from local friends?
