In this paper, we propose a novel approach to Bayesian Experimental Design (BED) for non-exchangeable data that formulates it as risk-sensitive policy optimization. We develop the Inside-Out SMC\textsuperscript{2} algorithm that uses a nested sequential Monte Carlo (SMC) estimator of the expected information gain and embeds it into a particle Markov chain Monte Carlo (pMCMC) framework to perform gradient-based policy optimization. This is in contrast to recent approaches that rely on biased estimators of the expected information gain (EIG) to amortize the cost of experiments by learning a design policy in advance. Numerical validation on a set of dynamical systems showcases the efficacy of our method in comparison to other state-of-the-art strategies.