It comes close on an every other day cycle, and if you waited to run it until filling it, it would probably beat that. (Maybe it goes without saying, but your ~1 gallon figure is extremely efficient for 15 minutes of hand-washing.)

That said, unless you live in a desert, or off-grid, 1-2 gallons of water a day has no real cost. E.g., I pay 0.7 cents/gal for the water and 2 cents/gal for the outgoing sewage.