A numerical approach for a time-optimal feedback control problem for an advection-reaction-diffusion model is considered. Our approach is composed by three main building blocks: approximation of the abstract system dynamics, feedback computation based on dynamic programming and state observation. For the approximation of the abstract system dynamics, we consider a finite element semi-discretization in space, leading to a large-scale dynamical system, whose dimension is reduced by means of a Balanced Truncation algorithm. Next, we apply the dynamic programming principle over the reduced dynamics, and characterize the value function of the optimal control problem in terms of viscosity solutions of the resulting Hamilton-Jacobi-Bellman equation, which is solved via a semi-Lagrangian scheme. Finally, the computation of corresponding feedback controls and its insertion into the control loop is performed by implementing a Luenberger observer.