I understand you have an array of pins that connect two plates together and must support an internal pressure.
You don't need to apply any pressure on the outside faces of the structure if you apply only the pressure difference to the inside faces of the structure.
I agree that you can slice through the centerplane of the structure because of symmetry, so you should apply a symmetry BC on that face, don't make it a Fixed Support.
The real structure has a limited extent, and the way that structure is supported can affect the stress. But I understand you want to simulate a unit of the plate assuming an infinite plate. CFD models have Periodic BCs, but I don't think that is available in Structural models, so I suggest you apply two symmetric BCs on adjacent faces by creating a coordinate system at a 120 degree angle so you can have two adjacent faces each with a displacement support that sets the normal displacement to the face at zero and leaves the other displacements free. That way you will have three planes each with a zero normal displacement and you will not need a Fixed Support. That should clean up the singularity caused by the Fixed Support.