Alright, I still don't have a Wii U to test with, but I now have a Family edition Wii to test the 3-inch disc adapter with, and it does make GC games work. Popped in Sonic Heroes, the disc was detected normally, and the game's demo loop has been running for 30 minutes. No strange sounds from the drive or anything. It just seems to work.
And no, since I haven't rigged any controller ports into the system, I can't actually play the game. I'm tempted to add a controller port in now though.
Hopefully this successful Family edition test encourages someone with a Wii U to test a disc adapter. There's been a lot of negativity and misinformation surrounding the Family edition's ability to read GC discs since that Wii model was released, so it would be nice to find out if the Wii U's vWii mode supports GC discs before the same type of misinformation leads people to believe it's impossible when it simply hasn't been tested yet.