While you technically chose the correct forum to post this in, unfortunately, this (the Alaska destination) forum is not super highly trafficked. If you're still looking for something, post back here and I'll see if I can get some better visibility for this post in other places where people who fly to/from ANC frequently may see it.

If you have any personal connections in a city served by Alaska Airlines, another option would be to have them ship it as a Goldstreak package (which goes out the next Alaska flight). Given the frequency of flights between Seattle and Anchorage, if someone in Seattle were to drop a small parcel with a Chick-Fil-A order in it at the Alaska cargo counter, it would very likely be available for pick-up at the Alaska cargo counter in Anchorage within approximately five to six hours. (You can also ship Goldstreak from anywhere else Alaska flies, but the transit time would be longer due to longer flights and/or connections.)

Delta also has a similar product called Delta Dash, but the few times I've checked into it, it's been more expensive, and they have fewer flights to ANC than Alaska.