This is just cursed. If you spend an entire year as someone else and living their life you will practically become that person. And doing so several times will really tarnish your self-identity.
How difficult would it be to just call Izuki's parents or something similar and just get her number? They are childhood friends so their parents must know of each other as well. Things are always so unnecessarily convoluted in these stories.
@SparkShadow the cashier only gave them the discount because they said they were a male and a female. If they had said that they were a girl they would not get the discount.