OK, here are a few questions nobody's asked:
Are you planning on having a sag car with your stuff behind you, or taking your own panniers? If you are taking your stuff along in panniers, rent a bike! What bike choices you'll get at Wally World will be cr@p. What you'll get in choices of racks won't be able to hold more than spit. If you're into performance cycling, rent or take your own - your knees will bless you. Spoken by someone who had one of hers rebuilt.
Rentals, for long-term, can be very reasonable. Add up the cost of the shipping container for the return flight, the hassle of transporting the boxed bike to the airport (many taxis won't take one), and the hassle of getting a bike from a retailer which isn't a bike shop (WallyWorld has anyone who can hold a wrench assemble their bikes. If they do it wrong you'll be broken down somewhere, or even hurting mercilessly). Simply avoiding the hassle of buying a cheap bike can be worth it. Try to get a multi-day discount on the rental. If you'd like to make a donation to charity, you could do it in cash upon your return, enough to purchase someone a bike, if that's your ultimate reason for donating the bike.
Remember, you'll either want to take or buy/borrow the basics in tools for repairs on the road - at the very least a flat repair kit, a set of wrenches, and perhaps a spoke wrench or a multi-tool. You'll need to check the bag carrying them, so if you have a bag which will hold your helmet (take your own - rental helmets are abysmally unconfortable!), pack it all together and consider excess valuation when checking it in (at the ticket counter, it's extra don't-lose-this-bag insurance). That tag, once put on, makes that bag radioactive to all baggage handlers. It will get where you need it, on the flight it's supposed to be on, probably in the correct container in the hold.
These days, you'll need to provide your own shipping box to take a bike. Check with your local shop to see if they can provide one; they can also tell you how to break down a bike to fit inside, and the tools you will need to have to do it. At the very least you'll need to turn the bars, remove the pedals, and perhaps the wheels and/or fenders. If you take your own bike, don't let it leave your hands without paying for excess valuation on it!
(In case you were wondering, I was an avid cyclist before one too many knee and back injuries sidelined me. Alas, my custom-built Trek, dusty and still loved but unused - bought l-o-n-g before Lance Armstrong probably ever learned the company existed!)
