We consider a Bayesian adaptive optimal stochastic control problem where a hidden static signal has a non-separable influence on the drift of a noisy observation. Being allowed to control the specific form of this dependence, we aim at optimising a cost functional depending on the posterior distribution of the hidden signal. Our setup is in sharp contrast to existing work: we include costs that depend on the full posterior distribution in a form that admits a large class of non-linear relationships. Expressing the dynamics of this posterior distribution in the observation filtration, we embed our problem into a genuinely infinite-dimensional stochastic control problem using measure-valued martingales. We address this problem by use of viscosity theory and approximation arguments. Specifically, we show equivalence to a corresponding weak formulation, characterise the optimal value of the problem in terms of the unique continuous viscosity solution of an associated HJB equation, and construct a piecewise constant and arbitrarily-close-to-optimal control to our main problem of study. As a byproduct of our analysis, we also provide a novel stability result for a class of measure-valued SDEs which we believe is of independent interest.