Would it be permitted to pray in a place where there are services that are mixed with Jews and gentiles? Meaning, a synagogue that also invites gentiles to pray with the Jews (but they all pray from a Sidur).
I've never heard of a synagogue that checks whether people coming in are Jewish or not, turning away those who aren't, so allowing Jews and gentiles to pray together is presumably not forbidden, and therefore, entering a building in which this happens is presumably also not forbidden.
In fact, given that gentiles were invited to bring certain kinds of offerings to the Temple, and entering the Temple certainly wasn't forbidden (at least, not for that reason!), it would stand to reason that synagogues that behave similarly would be treated similarly.