We'll be shipping UPS from Philadelphia to Missoula next month (LBS to LBS) for a loop tour. Did the same two years ago for a loop out of Whitefish. Using a LBS for the shipping may save you as I believe they get commercial rates. (Shipping to Whitefish via UPS Ground was under $50/bike each way.) The LBS in Missoula will assemble the bikes (at a charge) and hold the boxes and duffel bag for the gear, which will accompany us on the flight. At the end of the trip, we will simply ride to the shop, drop the bikes off, and have the LBS ship them back to our local LBS. The assembly and packing costs extra, but the convenience is worth it to me.
As for boxes, I have Pro XL II from Crateworks. Stronger than a cardboard box, yet less expensive ($179) than a hard case. And it's big, but not too big that you incur abnormal fees. I can get my 60 cm bike along with my racks (not attached) and a stuff sack containing my stove and fuel bottle in it. When I used one for a trip in Spain, I also got my sleeping bag and tent in it, too. I have used my current one for 4 round trip flights and it’s still going strong.
As noted, Frontier and Soutwest have manageable bike fees of $50 (compare to Delta, which I believe is over $200. Maybe as high as $275.) Unless you are flying either of those, you might find it more economical to ship via UPS or FedEx. We left 9 business days shipping to Whitefish to be on the safe side. I recall the actual transit time to be more like 6 or 7.
If you use an LBS for packing and assembly, check well in advance as to whether you need an appointment. Some are very busy during certain times of the year.
Amtrak is another very low cost option. You can purchase an Amtrak box for something like $12. It’s big and requires very little disassembly of the bike. I believe the fee for the actual shipping (if you are a train passenger) is only $15. The limitation with Amtrak is that, unless there is roll-on roll-off bike service, which is available on only a very limited number of routes, you can only check bikes as baggage between stations that offer checked baggage service. Such stations are often limited on many/most routes.) Not sure if you can still ship via Amtrak if you will not be riding the train. I know you used to be able to. Check Amtrak’s web site for details.