If the rotating pie is a pie that when nonrotating had the same radius as the other one, when it rotates it has a slightly larger radius (and circumference) because of centrifugal forces. This effect completely dominates over any relativistic one.
The centrifugal force can be arbitrary small. Say that we have only the outer rim of the pie, but as large as a galaxy. The centrifugal force at the half of speed of light is just negligible. Far less than all the everyday centrifugal forces we deal with.
Now say, that the rim has a zero around velocity at first and we are watching it from the centrer. Gradually, say in a million years time, it accelerates to a relativistic speed. The forces associated are a millionth of Newton per kilogram of mass. No big deal.
The problem is only this - where's the Lorent...
As per a recent comment this thread is meant to voice contrarian opinions, that is anything this community tends not to agree with. Thus I ask you to post your contrarian views and upvote anything you do not agree with based on personal beliefs. Spam and trolling still needs to be downvoted.