We develop a method for computing the stochastic wave speed of pulse solutions in kinematic equations subject to small stochastic forcing based on the isochronal phase reduction. These kinematic equations arise as the singular limit of sharp pulse solutions in the FitzHugh-Nagumo system, and our approach contributes a new perspective and method to the growing body of work on stochastic wave propagation in excitable media. The method yields an effective Itô process for the wave's position. The coefficients of the Itô process can be computed deterministically allowing for efficient computation. We demonstrate the efficiency and accuracy of our method through numerical demonstrations.