69a Daughter of a Jewish woman and a non-Jewish man or Eved to marry a Kohen

Yevamos 45a says vehilchisa goy v'eved haba al bas yisrael havlad kasher - that it is not a mamzer. But is this child completely kosher, or if it is a girl, is she unfit to marry a kohen?
There are three main opinions on this matter. Many rishonim pasken she may not marry a kohen. This gemara which says havlad kasher means she is not a mamzeres, but she is not completely kosher either, as Ravina says on daf 23a. For example, see Rosh, Yevamos 4:30.

Rif 15a holds the matter is a safek (doubt) if the daughter may marry a kohen.

Rambam (Issurei Biah 15:3) holds the daughter of such a union is completely kosher and she may marry a kohen. The Rambam just says kasher, but Beis Yosef EH 4 says since that's all the Rambam writes, he just means completely kosher. And everyone (except Mishna Lamelech) understands the Rambam this way.

Shulchan Aruch EH 4:19 says like the first opinion we mentioned: If [a non-Jew or eved] had relations with a Jewish woman, whether she was married [to a Jewish man] or unmarried, the child is kasher (see the context, that it means non a mamzer) but is pagum (unfit) for a kohen. This SA is a repeatition of how he paskened in 4:5. Beis Shmuel 2 says since the Rambam says she's mutar, and the Rif has a safek, if this daughter married a kohen we would not force them to divorce, and the children would be safek chalalim. This is based on the famous principle that the Mechaber usually paskens like the majority of the opinions of the Rosh, Rif and Rambam, and here each holds a separate way. However, Be'er Hagolah explains that according to the Shulchan Aruch, we would force them to divorce, and their children would be chalalim.