If a couple marries in a civil ceremony, their children will be considered "mamzerim".

Incorrect! Unlike the term "bastard" in English, a mamzer is not any child born out of wedlock. A mamzer (plural: mamzerim) is a Halakhic status applied to children conceived through a specific list of biblically forbidden sexual relations. The most common instances of mamzerim are children born from the extramarital affair of a halakhically married woman and a man who is not her husband. A mamzer and his offspring can only marry other mamzerim, and the status is inherited by all future generations. A child born to a couple married in a civil ceremony is not considered a mamzer and is not restricted as to whom he or she can marry.