Hope you don't mind one of the guys commenting, but I'm from Fairbanks myself, and I have flown several times in and out of the state, most recently outside the country (and making a return trip in a few weeks for dr's appointments). I'm not sure if you are flying out of FAI or through one of the bush airports, but if you are going through FAI TSA, you don't have much to worry about. Even Washington they didn't give me much trouble.
I would recommend, if you are seeing a therapist and presenting female that you have a letter stating that you are transitioning. I went through the full body scanners with my packer and binding, with my ID still saying female and I didn't get a single comment. TSA is slowly getting better about it. I was even able to take my hormones, needles and all through in my carry on without anyone saying anything.