Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with flux functions driven by low-regularity paths. For a convex flux, it is demonstrated that driving path oscillations may lead to "cancellations" in the solution. Making use of this property, we show that for alpha-Holder continuous paths the convergence rate of the numerical methods can improve from O(COST-gamma), for some gamma is an element of [alpha/(12 - 8 alpha), alpha/(10 - 6 alpha)], with alpha is an element of (0, 1), to O(COST-min(1/4,alpha/2)). Numerical examples support the theoretical results.