I understand that traditionally this is prohibited, but since they are the same faith would they require a ketubah or a modified version?
I read that Rav Feinstein would rather à kohen marry a Goya than a giyoret, so it got me wondering if they could write a ketubah:
Furthermore, Rabbi Moshe Feinstein (Responsa Iggerot Moshe Even Ha-Ezer vol. 2, #4) disputes Rabbi Hoffman’s assertion that by permitting a non-Jewish woman convert to stay married (civilly) to a kohen would allow for less transgression of the law. Rabbi Feinstein states that if the woman is converted and the couple does not follow the laws of family purity, the punishment is karet (divine excision) and the kohen violates the negative commandment (lav) of living with a zonah. However, leaving the woman unconverted involves only two negative commandments: not marrying a non-Jew and a kohen living with a zonah. Thus, Rabbi Feinstein rules that converting her makes the situation worse and he would advise doing nothing.