Shalom everyone, I'm just curious, if a Jewish man is dating a woman who is converting I assume it wouldn't be appropriate to serve on her beis din, but is it prohibited halachically?
Mishna Yevamot 2.10
מיאנה או שחלצה בפניו יישאנה מפני שהוא בית דין
The context of this end of chapter is that a man who is the cause of the permission to marry of a woman cannot marry her because of suspicion. People will say he was not objective as witness or as expert to cancel her vow, because he intended to marry her.
But if he was a part of a bet din, there is no problem because we have no reason to think that the other dayanim did help him making a sin. It's not plausible that two dayanim did make a sin because of one. The Tosfot Yom Tov quoted this statement from the Rif who himself quoted it from Yerushalmi.
The case of the question is worst. But bediavad, it seems that by the logic of the Yerushalmi it's valid because he was not alone.
We need to explore a possible possibility of shochad. Shulchan Aruch CM 9.2. rules that shochad devarim bediavad doesn't disqualifie a judge. Since this woman didn't give him a gift to judge her able to convert there is obviously no more (if there is) than shochad devarim.
If he was dating this woman before her conversion and marry her afterward this leads people to think that she converted because she wanted to marry him. But even if it was the case the Gemara Yevamot 24a (SA EH 11.5) ruled that the conversion is correct. So why is said in Mishna that if people was saying he has a relationship with a non Jewish woman he cannot marry her Lechatehilla? Gemara answers because of a recommendation in Mishle to avoid to provoke the rumor.