I have several DIY Colorado elk bow hunts with air travel (SWA). I would pack my take down recurve in a bow sock, limbs taped together in 1/2 of sock and riser in other half of sock. My 8 arrows would travel in an Eastern arrow tube. Both bow and arrow tube would be centered in my (BPS) 32 x 15 x 15" duffle bag and surrounded by clothes and gear as checked baggage. My camo backpack would be my carry-on bag.
We would take a 60qt wheeled Igloo cooler and pack our tent(s), tarp, cook gear, utensils, tools, etc and an insulated bag cooler (California Inovations) for camp use.
When we harvest an elk we would have it butchered, frozen and packed in our cooler and waxed cardboard boxes (with dry ice) for travel home as checked luggage. Also would take 45# meat in our insulated bag cooler as a carry-on.
BE Sure To Keep All Bags and Coolers Below 50#.
We would ship all our gear home with 5-7 day ground shipping.
We found this to be the easiest, cheapest, and most efficient travel.