Geodesic current supported on a pencil?

No - this is impossible because 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").

EDIT Following Lee Mosher's request I am adding more details concerning the definition of marginals in this situation. The marginals of a probability measure on a product space are its coordinate projections. However, geodesic currents are infinite measures, and just taking their coordinate projections would lead to trivially infinite measures (or, at least, to measures which are not Radon for discrete currents). What I meant was not marginals themselves, but rather their measure classes, which make perfect sense. Namely, one should take a finite reference measure $\lambda$ equivalent to the geodesic current; then the measure classes of its coordinate projections do not depend on the choice of $\lambda$.