Its hard enough trying to get it on a company flight, let alone our mainline affiliates, but on another carrier....good luck
If you can list ahead of time that will likely help you get in the system, but I'd love to see the ticket agent's face when you roll up with a board bag.
You: Hi, I'd like to list for the jumpseat please
CS: I don't know how to do that/We don't have CASS here/Who do you work for?!?!
You: Oh and by the way, I'd like to check this.....item
I'd say your best and cheapest bet is on a company flight. If you're originating at a SkyWest station they may not charge you, but if its United or Delta, you'll likely pay anywhere from $125 to $300 (A bad personal experience with Delta in a foreign country).
Either way, good luck and enjoy your new stick!