Physics-informed neural networks (PINNs) offer a mesh-free approach for PDE-governed mechanics. For plate bending, existing PINN studies have largely emphasized deflection-level validation, while mechanics-consistent assessment of in-plane stresses and membrane stress resultants remains limited for geometrically nonlinear thin plates governed by the Föppl--von Kármán (FvK) theory. This limitation is practically important because many engineering decisions for thin-walled structures are driven by stress- or force-based criteria rather than deflection alone. This work addresses this gap by developing a Fourier-enhanced, consistently nondimensionalized PINN for large-deflection FvK plates with explicit stress-resultant verification. A mixed formulation is adopted, where the network predicts the transverse deflection $w(x,y)$ and an Airy stress function $F(x,y)$. The distributed transverse loading is denoted by $q(x,y)$ and incorporated via the strong-form residual $R_w=\mathcal{L}_w(w,F)-q$, together with the compatibility residual $R_F=\mathcal{L}_F(w,F)$. Coordinates, outputs, and differential operators are nondimensionalized to balance multi-term nonlinear residuals across geometric and material scales, improving training robustness in parametric settings. Fourier features are embedded at the input to enhance the representation of high-frequency and boundary-layer responses, which are frequently associated with large-deflection plate solutions and are critical to reliable recovery of in-plane quantities. A simply supported square plate is selected as the benchmark. Verification is performed using manufactured solutions providing closed-form $(w,F)$ and the corresponding $q(x,y)$, enabling quantitative assessment of both $w$ and in-plane stresses and membrane stress resultants derived from $F$ (with $\sigma_x=F_{yy}$, $\sigma_y=F_{xx}$, $\tau_{xy}=-F_{xy}$ and $N=t\sigma$ under the adopted convention). We report squared relative errors $\mathrm{SRE}(w)=\|w-w^\ast\|_2^2/\|w^\ast\|_2^2=2.2\times10^{-6}$ and $\mathrm{SRE}(N)=\|N-N^\ast\|_2^2/\|N^\ast\|_2^2=5.6\times10^{-4}$ (for $N_x$, $N_y$, and $N_{xy}$). In addition, a uniformly loaded simply supported square plate will be investigated as a further validation to demonstrate nonlinear load--deflection behavior and membrane stress-resultant field prediction against high-fidelity finite element simulations.