With respect to the THOUGHTS ON THE DAILY DAF (http://www.shemayisrael.co.il/dafyomi2/bechoros/insites/be-dt-47.htm)
The NESIVOS HA'MISHPAT (Tshuvos ha'Gaon mi'Lisa #25) proposes an even more unusual solution. Perhaps the rule that "a non-Jew who converts is considered to be newly born, with no ancestry" does not apply in its full sense to a child born to a gentile by a Jewish woman. Once he converts, he is retroactively considered to be the son of his Jewish mother. Therefore, if his mother is a Levi'ah he, too, will be a Levi!
1) If the Father is a Levi and the mother is Jewish but not Leviah are here sons Levi and her daughters Leviah?
2) If the Daughters are Leviah and one of the daughters marries a Jew who is not a Levi are her sons Levi?
Maazya, Houston, Texas U.S.A.
2) With regard to the laws of Pidyon ha'Ben (redemption of the firstborn son), if one parent is a Levi -- even the mother -- then there is no requirement of Pidyon ha'Ben. For all other Halachos, the status of the child follow the status of the father only.