I never even thought of the paper cuts tbh, I initially thought the joke was that he fell all the way to the bottom of the pool and bruised himself since dollar bills wouldn't be good at cushioning the impact.
To be fair, that is probably the most fitting bikini she could wear. It shows off how rich she is (which she loves to do) and also shows off her body (which she doesn't like to do, but often unintentionally does anyway).