If a woman converts to Judaism while pregnant, will her child be eligible to marry a Kohen?

Question

If a woman converts to Judaism while pregnant, will her child be eligible to marry a Kohen?

Case #1 -- Assume her husband is a Jew by birth.

Case #2 -- Assume her husband converted to Judaism prior to the child's conception. (I know the kohanim have assumed an extra stringency against marrying the child of two converts. However, would there be a Biblical prohibition in the case I describe?)

Case #3 -- Assume her husband was a non-Jew at the time of conception. (Again, would there be a Biblical prohibition? --I know there would be rabbinical prohibitions as discussed here.)

Answer 1

The Rambam writes (Issurei Biah 19:12) regarding the case of two converts who married each other and had a child who went and married a kohein:

ואם נשאת: לא תצא, הואיל והורתה ולידתה בקדושה
And if they did get married, they need not get divorced since her conception and birth were in holiness [ie. of a Jewish mother].

This implies strongly that if both the conception and the birth were not "in Holiness," than the child would be forced to divorce her kohein husband, the general indicator of a biblically forbidden marriage. This understanding of the Rambam is found in both the Beit Shemuel (sk 42) and the Chelkat Mechokek (sk 29) to Shulchan Aruch Even HaEzer Siman 7.

It seems they are all agreeing that this child is considered a full-fledged convert in her own right, which would make her biblically forbidden to a kohein independent of her father's status. The other discussions you reference in your question are talking about cases where the mother was already Jewish when the child was conceived.