Talk to the airline cargo sales offices directly - they are the ones who know what kind of load factors (both by mass and by cubic size) their flights are running, and they are the ones who could advise you whether or not you could ship the bike uncrated at a cost comparable to a crated bike.

Keep in mind that a crate will add weight. If the carrier is not running physically full freight holds (meaning, if they are not cubed out), it won't matter to them if it is crated or not. But weight always matters. In other words, it's not uncommon for an airline to waive cubic charges to get business, but no airline will ever waive weight charges.

Plus, when you get into crating, you encounter other DG related requirements such as labelling, concerns about the battery and fuel level, etc. that you don't encounter when the moto is shipped upright and fully assembled. And there is a whole set of rules about the kind of wood needed for the crate - it has to be specially labelled and specially treated to ensure that insects & fungi are not accidentally transported because they had previously infested the tree the crate was made from.

I would try to avoid crating unless there is no other alternative. If you have to go the crating route, I strongly recommend you engage an agent who is familiar with transporting motorcycles by air.

Michael