Why is an electronic circuit system should be linear to use Fourier and Laplace transforms?
Because a nonlinear system doesn't benefit from the superposition property, that is essential to use the transforms. Another property that is not valid for nonlinear system is the homogeneity.

The only solution is find the equilibrium point, because for little excursions from this point you have a linear system, and you can approximate the behavior of the system by this way.