Check with the airline. I am planning to fly Southwest this Fall with my bike. Their policy on the web was pretty vague and kind of self-contradictory but then my riding buddy went to the airport ticket counter and got the straight poop on what the actually do when confronted with a bike. I'm not sure Southwest would accept a bike that was not in a box.
I think a hard case is a good idea. If you get where you are going and your bike is seriously damaged your entire tour is likely to be in trouble. Even if money were no object, having to go out and get some repairs done while you wait or worse still buy a replacement is likely to cost a few days. And presumably you're doing this not in your home turf.
I had a bike damaged during air transport once but fortunately it was on the return leg and it was not fatal. I had broken down my bike and not boxed it according to the airline's instructions. The handlebars were turned parallel to the frame, the front fork turned backwards, the pedals threaded into the cranks from the insides, the wheels attached to the frame, blocks fastened into the front and rear dropouts, etc. Near as I could tell, somebody dropped the bike onto the crank arm. The drop was significant enough to bend the crank axle slightly and crunch a pedal. I replaced the pedals and I rode the bike afterwards for many years but the orbital plane of the chainrings was never quite flat after that. If it had been much worse and if it had been on the departure leg of my trip it could have been really unpleasant to deal with.
I bought a case from Nashbar. I think it is still on sale. If your airline will accept it, I think shipping your bike naked may definitely be a better strategy than a flimsy cardboard bike box but I'm putting my money on the cargo case for this Fall's trip.