We extend the Rate-Distortion-Perception (RDP) framework to the Rényi information-theoretic regime, utilizing Sibson's $α$-mutual information to characterize the fundamental limits under distortion and perception constraints. For scalar Gaussian sources, we derive closed-form expressions for the Rényi RDP function, showing that the perception constraint induces a feasible interval for the reproduction variance. Furthermore, we establish a Rényi-generalized version of the Strong Functional Representation Lemma. Our analysis reveals a phase transition in the complexity of optimal functional representations: for $0.5<α< 1$, the coding cost is bounded by the $α$-divergence of order $α+1$, necessitating a codebook with heavy-tailed polynomial decay; conversely, for $α> 1$, the representation collapses to one with finite support, offering new insights into the compression of shared randomness under generalized notions of mutual information.