American Flag on pole, topped with an Eagle. Is this permitted in the Synagogue?
Rabbi Moshe Feinstein wrote that US and Israeli flags inside the synagogue (i.e. the worship room itself) were permitted; there was nothing idolatrous about them. He felt they were less-than-ideal, simply as the prayer room should be dedicated only to prayer. He stressed it was far better to pray in a dedicated synagogue, even if it had flags, than in someone's basement.
He did not address the finial per se, but I'm going to make a reasonable guess here -- most of the synagogues I've seen that have flags (whether in the sanctuary or the lobby) have had eagle finials. I'm pretty sure that was the standard practice at the time Rabbi Feinstein wrote his responsum. So when he said "it's not idolatrous to have a US flag in the sanctuary", I think that means "even if it has the little eagle thingy on top." Provided no one is actually worshiping the thing... otherwise we have bigger issues...