I knew the center hit would be pure translation, so I started solving for the end strike. It's definitely not pure rotation.

I think I just figured out the trick! Treat the rod as two point masses at the ends of a massless rod. The mass that is struck is at that instant taking the full force. Solve for that motion, and the other end of the rod is motionless.

The general case formula is still a nightmare, but I can solve for the two extreme cases.