CA, I do not say anything. Rav Yosef Karo says it based on Rabbi Tarfon (Kiddushin 69a). See also Bet Shmuel and Chlekat Mechokek thatt his was allowed for the sake of the child – which means that here is no need for him to become an eved Ivri. Moreover, the Mechaber does not include any halchot that are not in force in our time and until Yovel is re-established there cannot be an eved Ivri (Rambam, Hilchot Avadim 1:10).You say? Do you have broad enough shoulders to disagree with the above? If not, cite someone who does.
In any case, poskim go to great lengths to clear someone from mamzerut. I heard that once Rav Moshe discovered that a BT’s mother was under psychiatric care when she married her first husband (who did not give her a get). He went to the psychiatrist personally and told him that he had only lived so long in order to clear the girl (the psychiatrist was by then quite elderly). The info allowed Rav Moshe to rule that the mother was halachically a shottah at the time of her first marriage. Similarly, it is well-known that Rav Moshe was meikal regarding women who had been married in non-Orthodox ceremonies. Possel eidim also occur. Once someone told me that before he became observant he was a witness at his sister’s wedding.