Here are a few frames showing Step 2, the transient portion where there are no changes in the Boundary Conditions.
At the end of Step 1, the structure is compressed. The contour is the X displacement.
As I said before, if you extrude a deeper structure, 10 or 15 mm instead of only 5 mm deep, that may help to stabilize the structure. You could also increase the thickness of the horizontal springs to reduce the chance of buckling. This is an actual physics result, not an artifact of the simulation.
I increased the horizontal spring by 5% to 0.84 mm and that seems to have fixed the problem. I could get through step 2 with only 100 initial substeps.