Does anybody know, if klesis was ever used by the ancients as a call to salvation like in the NT? And, if not, what was the dominant meaning of this word in the Classical Greek? I would really appreciate any insights on this point.

Well, the PASSOW quotes klesis = "Anrufung (call) of the GODS, Menad. rhet. p. 132, 10 Walz."
It, obviously, refer to the Walz's edition, that I don't have, on the WEB is freely available the Spengel's Edition of the RHETORES GRAECI with some of Menander' works.