Thanks for the suggestion! I gave it a shot, but I get 403: Forbidden with that app (and the techniques that the other person suggested).
EDIT: Actually I take it back-- The iframe version of the URLs will 403, but the normal versions (/watch/ instead of /ifr/) work fine
Okay that's been working SUPER well for me-- Thank you for laying that out. Seriously a game-changer, it's actually better than what I was doing before that stopped working, lol.