I've shipped my bike many times with the Trico Iron Case. Easy to pack, protects the bike, good product.
When I first starting shipping it, east coast to midwest was $40 by Fedex. Now it's $150+. Turns out the Iron Case is just slightly oversize and Fedex cracked down on this a few years ago.
Most bike boxes, even cardboard, will be oversize. Bikeflights.com can probably help you with this.
I've never shipped a bike by airline. I don't want TSA unpacking/repacking my bike, and the airlines will pay you pennies on the pound if they damage or lose the bike.
With Fedex you can insure to whatever amount you want. (I recently sold a carbon wheel on ebay, shipped it fedex, and got paid for the insured amount after the wheel arrived damaged. I insured it for just a bit more than the auction amount; should have insured it for replacement retail, over 3X my insured amount...)
Bikeflights, by the way, handles insurance just a bit differently, and I haven't sorted that out yet.
Or just rent a bike...or grab a Lime.
