My first suggestion is to take the time to learn to cape your turkeys for mounting. It isn't difficult and it makes your conundrum about flying with a bird much, much simpler. First thing first though, whatever you are planning on doing, you've got to keep the bird or cape as cold as possible (preferably frozen) from the time you kill it until your flight. Not doing so may result in the cape spoiling,...in which case you are screwed.
I have flown with caped birds both in my carry-on and in checked baggage. Note: you will get some very strange looks from the folks at the airport when you run a carry-on with a turkey through the x-ray machine.
If you just don't have the confidence to cape your bird for flying, the best thing to do is to freeze the entire bird (or otherwise keep it as cold as possible) and put it in your checked baggage or carry-on. Obviously, you would have to have room for it in your existing baggage or have an additional bag with you to put your gobbler in. For a whole bird, I would probably go the checked-baggage route in that the carcass will stay much colder that way (temps in the cargo hold at 35,000 ft. are pretty cold from what I hear...
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