Assuming this few hundred dollar rule is the case, even more of reason to use a hardshell bike box.
On three trips I've used a Trico Iron case. Needless to say, I don't worry too much about my bike arriving in one piece. It's all a matter of personal preference. Your best bet is to book a night or two on both ends of your trip with the same hotel/hostel. All of the times I've flown with my bike the hotel has always agreed to keep my case for no extra charge. A email or two will find out if they'll do this. Many chain and 3-4 star plus hotels offer pick up and drop off at the airport. A beautiful thing after a long flight. There are some deals if you look around. A few years ago in The Netherlands two of us scored a room in the Raddison for $90 a night for the first and last night we were there. They kept our bike boxes as part of deal. I leave any extra travel clothes or other items I don't want to take along on the bike trip in the case. Obviously, if you're not flying out of the same airport you fly into, a cardboard box is probalby you best option.
Good luck, and let us know how it works out!