Try a multistep solution sequence.
Step 1. Apply pressure to the non-facesheet panel walls to create some small deformation to seed the buckling.
Step 2. Apply some of the axial load to the facesheet to "hold" the small deformation from step 1 from leaving.
Step 3. Ramp on some more axial load while ramping off the pressure load to zero.
Step 4. Apply the remainder of the axial load.
I suggest alternating the direction of pressure in adjacent cells.
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