No. The Double Plot Prizes Fortune Cookie does not stack with the Double Plot Prizes Premium Perk, and therefore you cannot earn 4x the prizes.
You will not be able to use the Double Plot Prizes Fortune Cookie at all if you have a Premium Subscription (though you can still purchase one for trading purposes, or if you decide to cancel your subscription later on).
If you have a Premium Subscription, you will automatically get double the plot prizes every time you earn them.