You don't need to request cashout for this.
Simply make a daily bankroll update and put the amount next to the new poker site and put 0 for the older one.
We understand it can take a day or a few days until the money is "really" on the other site. But don't worry, in order to make accounting easier, pretend it is already there. It's not like you will lose the money, so the account is still correct and the correct profit number will show up.