Apparently this is a real chore, and even more so if you're using alternate launchers like SKSE and/or Mod Organizer. So far, I think I've got it working with Mod Organizer by simply adding it to my Steam library, and then launching it from there in Big Picture mode.
Now I have my next problem though. The anolog stick controls movement, but the movement itself is still not anolog so it's just full speed ahead whenever I move the stick. This was pretty much the main reason for me using the controller with Skyrim in the first place, so this was really disheartening. Is there a way to make that work, or is it flat out not supported?
I tried ticking on the 360 pad feature in the options, but apparently that is a very bad idea in conjunction with a Steam controller profile and I pretty much lost control of the game at that point.