Its not how bitcoin works, but its an interesting concept. Instead of having the whole world verify blocks, one could somehow use nodes in some proximity. One could call it local web of trust.
One worry is the bar thinks its getting paid because it's seeing false confirmations, when in reality it's giving away drinks for free because the transactions are never getting relayed to the rest of the network.
Nope. A "confirmation" is a valid block with the transaction included, and a valid block includes a proof of work. The proof of work is far too valuable to waste trying to double spend in a bar.