Naturality of Heegaard Floer invariants under positive rational contact surgery

For a nullhomologous Legendrian knot in a closed contact 3-manifold Y we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of Y is mapped by a surgery cobordism to the contact invariant of the result of contact surgery. In addition we characterize the spin-c structure on the cobordism that induces the relevant map. As a consequence we determine necessary and sufficient conditions for the nonvanishing of the contact invariant after rational surgery when Y is the standard 3-sphere, generalizing previous results of Lisca-Stipsicz and Golla. In fact our methods allow direct calculation of the contact invariant in terms of the rational surgery mapping cone of Ozsv\'ath and Szab\'o. The proof involves a construction called reducible open book surgery, which reduces in special cases to the capping-off construction studied by Baldwin.