In layman's terms, the Casimir effect is an outside pressure pushing the plates together. It comes from modes of quantum fields that have longer wavelengths than the separation of the plates. Therefore, these modes can no longer be excited by vacuum fluctuations.
Since special relativity is a basic ingredient in QFT, these fields are homogeneous and isotropic. Thefore a constant Lorentz transformation will not alter the measurement of the Casimir effect.
In general relativity, we allow for non-constant Lorentz transformations and even for fully general diffeomorphisms. Then, there are new effects to consider, most prominently the Uhruh effect.
The derivations of the Unruh effect assumes constant acceleration for simplicity. A rotating frame implies non-constant acceleration for parts of your plates. A naive consideration in your thought experiment could go like
- Let us consider only the outermost edge of each plate
- These edges are accelerated towards the center of mass
- If we are allowed to consider infinitesimal time slices, the acceleration is approximately constant for any given time slice
- The acceleration towards the c.o.m. implies (in the rest frame of the edge) Unruh radiation, reducing the magnitude of the acceleration
- Therefore the measured attraction between the plates should be reduced
The above argument assumes that we are allowed to approximate the acceleration to be constant at each instant, which might not hold. Also, the Unruh radiation only exists in the rest frame of the edges, which are different frames for each edge, so concluding that there is less attraction from just considering one edge is quite bold. Finally, I did not carefully consider the transition from the rest frames to the lab frame.
All these caveats could be solved by careful consideration, but that is beyond the scope of my interest for this thought experiment :-P