Okay, after a bit of tinkering with it, here's what I've figured out.
I downloaded "New Super Mario Bros. U" using Wii U Helper. I also downloaded the update to get Pro Controllers to work. Wii U Helper also downloaded a version of Cemu 1.7.3d, so it should support updates. Whether I load Cemu through Wii U Helper or load it manually, it crashes at the title screen when pressing "A".
I manually downloaded a copy of Cemu 1.7.3d. If I start that one, and load the same version of the game that Wii U Helper installed, it does not recognize the update at all.
Is there a way for me to manually decrypt the update and apply it to the game? That's the only other solution that I can think of that might work. I think that Wii U Helper soft-updates the game (for lack of a better term, sort of how some emulators can soft-apply an IPS patch and let you play a hack without actually patching the ROM), and Cemu doesn't seem to like that very much.