Given a strictly convex set K\subset\rr^2 of class C2 we consider its associated sub-Finsler K-perimeter |\ptl E|_K in \hh^1 and the prescribed mean curvature functional |\ptl E|_K-\int_E f associated to a function f. Given a critical set for this functional, we prove that where the boundary of E is Euclidean lipschitz and intrinsic regular, the characteristic curves are of class C2. Moreover, this regularity is optimal. The result holds in particular when the boundary of E is of class $C^1