You mentioned: A Cohen can’t marry a convert even if converted as a baby. (Shulchan Aruch EH 6; 8) Picture this: A non-Jewish women dates a Cohen (no marriage). While she is converting a daughter is born. At the age of two, both the daugther and the mother finish their convertion and become Jewish. Now the daughter is dating a Cohen. What is the halagic status of the daugther? Can she be considered Jewish while her mother was already converting? Can there be a mariage?
According to the Rambam and the Shulchan Aruch, a Cohen may not marry a convert, even if the conversion was done as a baby. The case you describe does not significantly change those aspects. If the case is an actual one and not theoretical, I strongly adsvise consulting with a Dayan who is an expert on the laws of marriage, since there always may be a factor which one is unaware of.