Many real-world tasks require optimizing expensive black-box functions accessible only through noisy evaluations, a setting commonly addressed with Bayesian optimization (BO). While Bayesian neural networks (BNNs) have recently emerged as scalable alternatives to Gaussian Processes (GPs), traditional BNN-BO frameworks remain burdened by expensive posterior sampling and acquisition function optimization. In this work, we propose {VBO-MI} (Variational Bayesian Optimization with Mutual Information), a fully gradient-based BO framework that leverages recent advances in variational mutual information estimation. To enable end-to-end gradient flow, we employ an actor-critic architecture consisting of an {action-net} to navigate the input space and a {variational critic} to estimate information gain. This formulation effectively eliminates the traditional inner-loop acquisition optimization bottleneck, achieving up to a {$10^2 \times$ reduction in FLOPs} compared to BNN-BO baselines. We evaluate our method on a diverse suite of benchmarks, including high-dimensional synthetic functions and complex real-world tasks such as PDE optimization, the Lunar Lander control problem, and categorical Pest Control. Our experiments demonstrate that VBO-MI consistently provides the same or superior optimization performance and computational scalability over the baselines.