I've experimented with a couple of options but had no luck so far:
Option 1: C3 JS sends commands directly to the YouTube iFrame which is embedded in the C3 iFrame.
Option 2: C3 JS sends commands to the C3 iFrame, which then sends commands to YouTube iFrame.
Unfortunately I don't know if either of these options is technically possible, and if so, what code to use.
I've created a quick project to illustrate the problem and I'd be grateful if anyone can help with a solution (or put me out of my misery and tell me why it's not actually possible):