We study SDEs arising from limiting fluctuations in a $(2+1)$-dimensional surface growth model called the Whittaker driven particle system, which is believed to be in the anisotropic Kardar--Parisi--Zhang class. The main result of this paper proves an irrelevance of nonlinearity in the surface growth model in the continuum by weak convergence in a path space; the first instance of this irrelevance is obtained recently for this model in terms of the covariance functions along certain diverging characteristics. With the same limiting scheme, we prove that the derived SDEs converge in distribution to the additive stochastic heat equation in $C(\Bbb R_+,\mathcal S'(\Bbb R^2))$. The proof addresses the solutions as stochastic convolutions where the convolution structures are broken by discretization of the diverging characteristics.