I was wondering if there was something in any of the game's data that would allow me to carry over the values of my current character's face to a new character, without resorting to a mod?

Easiest method would be using Wrye Bash - just right click on the save you want to bring the face into - select import face and then select the save you want to get the face from and select it and it imports it for you !

That reminds me of the good ol' FaceGen standalone ^^ You can make better faces in FaceGen than you could ever create in the in-game FaceGen modeller, even your own face can be implanted with FaceGen's face recognition feature. Just pointing out, even though it is a little off-topic