Small genus knots in lens spaces have small bridge number

In a lens space X of order r a knot K representing an element of the fundamental group pi_1 X = Z/rZ of order s <= r contains a connected orientable surface S properly embedded in its exterior X-N(K) such that the boundary of S intersects the meridian of K minimally s times. Assume S has just one boundary component. Let g be the minimal genus of such surfaces for K, and assume s >= 4g-1. Then with respect to the genus one Heegaard splitting of X, K has bridge number at most 1.