I finished the study of using different constraints with the two possible static analysis solvers.
You can check out the project here.
In the basic static analysis i used the constraints that you already had plus an additional where i either additionally fixed DY or DX, doth gave the same result.
In the advanced static analysis I used a remote displacement BC fixing all 6 DOFs and put the reference point to the center (0,0,0), with activating the deformable option. So this constraint should not add any stiffness.
Here you can see the comparison of both (the essentially lead to the same results):
Displacement magnitude. Left: Static Analysis Right: Static Analysis - Advanced
Von Mises stress. Left: Static Analysis Right: Static Analysis - Advanced