Over the past dozen years I've done most every route with a kayak. One time I flew with 5 kayaks and one personal bag. Last year I flew with a creek boat, a rear lift-gate window to a Chevy Tahoe, 2 SUP boards, and large personal bag. It's like a fun challenge for me... but it's not for everyone.
Here are the details of my airline cargo experience. At the time I was the director of Huge Experiences (kayak high school). We sent 13 kayaks air cargo (think it was American Airlines). I recall it coming to about $115/boat which seemed like a lot back then and it was more than the expected $75/oversize bag. But we paid extra for the guarantee factor.
Myself and a solid Chilean logistician spent the next day in customs (lots of going back and forth and back again with varying papers). The group hungout at the Rio Maipo and were upset to not paddle that first day. After a sweaty day with customs they checked out.
What you will need to ship aircargo.
You need an on-the-ground Chile customs agent or logistics person to list on the papers. You'll need an additional day and a vehicle to go back and forth between offices. Was worth the effort for 13 but I think too much for one boat.
I now own Pucon Kayak Hostel and rent kayaks as a solution to this challenge. Still leading youth trips too... so lots of kayak transport happening. At approximately $US20/day many boaters spend less renting than if they'd pay in excess luggage fees. For extended rentals we'll work with you on a discount (especially if it's not peak-season).
Pucon Kayak Hostel Kayak Fleet
Ian Garcia is selling new kayaks at good rates in Pucon. Kayak Chile and Kayak Pucon also rent affordable kayaks.
If you decide to rent then I'd like the opportunity to give you a quote.