The equations don't care how quickly the process is done, only whether we add any energy. There are no "rate of change" terms in the gas laws. We are taking the gas in one tank and letting it expand into two.
You are adding energy to the receiving tank, that's why it gets hot. The donor cools off. Between both tanks and the whip there is no change in stored energy. The faster you fill, the colder the donor and the hotter the receiver will be. If you do it really slowly (days) the thermal isolation between the tanks provided by the transfill whip won't exist. And both tanks will remain at room temperature. Amazing huh?
So in the 0 psi tank going to 1500psi. What's that called?