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.