“That would only apply if he embarrasses someone when he is not allowed to (which is most of the time). If it’s a situation in which he is halachically required to, it would not apply. I believe there are situations in which a Beis Din may be required to embarrass someone publicly.”
I believe you should never embarrass someone without a constructive purpose. My proof is Bar Kamtza, an evil man, who was embarrassed at a party attended by the leading Rabbis of that time (who had greater minds then anyone in the coffee room) and resulted in the destruction of the second Temple. According to you the destruction should have been caused by something else.( Note there were other factors but Bar Kamtza was one of the main factors)
There is a big difference between Bais Din punishing someone and the person being embarrassed. There the intention was to punish the person not embarrass him. In this case they are trying to embarrass him as pressure to give a Get. Also is this social pressure sanctioned by Bais Din? Also at least here in the USA there is no obligation to follow one Bais Din so the husband can claim he is following a different Bais Din resulting in embarrassing an innocent man.
I am not saying it’s 100% prohibited to use social pressure in this case. What I am saying is that it’s a minefield, as you can lose your portion in the next world and should be avoided.