We derive Gaussian approximation bounds for $k$-Potential Nearest Neighbor ($k$-PNN) based random forest predictions based on a set of training points given by a Poisson process under fairly mild regularity assumptions on the data generating process. Our approach is based on the key observation that $k$-PNN based random forest predictions satisfy a certain geometric property called region-based stabilization. We also compare the rates with those of $k$-nearest neighbor-based random forests, highlighting a form of universality in our result. In the process of developing our results, we also establish a probabilistic result on multivariate Gaussian approximation bounds for general functionals of Poisson process that are region-based stabilizing. This general result makes use of the Malliavin-Stein method, and is potentially applicable to various related statistical problems.