Proximal splitting methods are standard tools for nonsmooth optimization. While primal-dual methods have become very popular in the last decade for their flexibility, primal methods may still be preferred for two reasons: acceleration schemes are more effective, and only a single stepsize is required. In this paper, we propose a primal proximal method derived from a three-operator splitting in a product space and accelerated with Anderson extrapolation. The proposed algorithm can activate smooth functions via their gradients, and allows for linear operators in nonsmooth functions. Numerical results show the good performance of our algorithm with respect to well-established modern optimization methods.