Since the problem is not time-dependent, I would not use dynamic analysis (modal, implicit, or explicit). The main non-linearity is geometrical, so a simple *STATIC card with NLGEOM would be sufficient. Be aware that the solution depends on the “way” traveled to the solution, so smaller steps are needed to achieve a meaningful result.
Since your flexure is composed of 4 beams fixed in the outer structure guided by the inner structure while subjected to identical moments due to symmetry, I think it would be nice to use the formulation from section 5.3 of Larry Howell’s book so that your model becomes pseudo-rigid links made stiff by torsional springs:
I would also refine your mesh in the flexure (it is very coarse at the moment).
You could compare analytical, simulation by FEM, and experiment. However, keep in mind that 3D-printed models are inherently anisotropic.