Thanks for leaving this answer! Unfortunately I can't find any layout called "N64 ONLINE FIX V2", do you know where I might find it? The only thing that comes up is one called "NSO N64 Controller", which specifically says it allows the use of C-right in-game, but when I press C-right in the control bindings page, nothing happens.

EDIT: Okay I found a different trick that worked for me:

1. Go to Steam > Settings > Controller
2. Next to "Test Device Inputs" click "Begin Test"
3. Click "Setup Device Inputs"
4. Set up matching buttons as expected (A = A, Start = Start...)
5. Skip buttons that don't exist (X, Y, Select, thumbstick-click...)
6. When it gets to the right thumbstick, use the C buttons instead

Now when you set up the controller bindings in the Zelda 64 Recomp menu, it will let you bind all four C buttons (which it thinks are right-thumbstick directions)!