The Rabbinate has no authority over me because our wedding ceremony was not conducted by a Rabbi or in any religious manner.

If the ceremony did not include any religious components, then this is true. However, if the ceremony included any religious elements that would render the couple halakhically married, then this is incorrect. Because the Rabbinate has exclusive jurisdiction when it comes to divorce in Israel, if one of the partners "tattles" to the Rabbinate that the couple was married, the Rabbinate can blacklist the couple as "ineligible for marriage" until they go through the requisite divorce process at the Rabbinic Court.