I do not want to piss on the parade here, but is the whole idea of the Lightning Network not to have a 100% digital payment network? we will go backwards, if we start the physical printing of "money" again.
The whole idea of Bitcoin, was to replace the old way of physically printing money and now people want to go back to that again? Nope, I think you must go back to the drawing board with your idea and eliminate the "printing" idea.

exactly my thoughts. why would you want to use bitcoin if you want to carry around a "piece of paper", just use cash, it is more convenient that way and it is accepted everywhere.
even if you wanted to use bitcoin then why use lightning network? why not simply use bitcoin with on-chain transactions? you can print private keys each containing a certain amount of bitcoin in them (10k, 50k,...) and go around giving them as money and if there is "change" you can also print an "address" on the same paper which they can send the change back to.