Two thoughts.
First, I'm also a bicycle rider. My friends and I often travel with our bikes in a box. Europe, Caribbean, Canada, etc. The typical case is 47 x 31 x 11 and fits on many airliners. While I know of no one that has had a problem, Howard has related at least one situation a stunt pilot encountered.
Second, using FedEx Ground, UPS or Greyhound also works. However, I'd advise estimating your dimensions and getting a shipping estimate on their websites. Don't worry about accurate weight, because you will be charged "dimensional weight," meaning the dimensions are the critical factor.
best of luck,
Peter