I have a similar problem with my nyckelharpa...it does NOT fit in the overhead bin. I have had a special case (one that was meant for a key board) to hold the wooden case. This is the only way i would transport it by plane..(by the way Arlo was taking a different type of KEYS)
The companies tha tmake cases for air line travel are:
flight case http://www.flightform.com/
Calton http://www.caltoncases.ab.ca/