We characterize the behavior of a random discrete interface $φ$ on $[-L,L]^d \cap \mathbb{Z}^d$ with energy $\sum V(Δφ(x))$ as $L \to \infty$, where $Δ$ is the discrete Laplacian and $V$ is a uniformly convex, symmetric, and smooth potential. The interface $φ$ is called the non-Gaussian membrane model. By analyzing the Helffer-Sjöstrand representation associated to $Δφ$, we provide a unified approach to continuous scaling limits of the rescaled and interpolated interface in dimensions $d=2,3$, Gaussian approximation in negative regularity spaces for all $d \geq 2$, and the infinite volume limit in $d \geq 5$. Our results generalize some of those of arXiv:1801.05663.