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No - this is impossible and follows from the fact that any ergodic geodesic current either has purely non-atomic marginals or corresponds to a closed geodesic. Indeed, the quoted result from Martelli implies that if one of the marginals contains atoms then it has to be concentrated on the orbit of an endpoint of a periodic geodesic (actually, this is true for a compact negatively curved manifold in any dimension without requiring that the curvature be constant). The Birkhoff ergodic theorem in combination with the asymptotic convergence of the geodesics with the same endpoint then implies that the invariant measure coresponding to this current is the one concentrated on the periodic geodesic (this is the so-called "Hopf argument").