I notice that in responding to a question about the 10 northern tribes, that you refer to them as being Yahudi, and the Hebrew religion, and heritage as being jewish. I have no real problem with that, although, in order for Mashiach to come, is it not necessary for all twelve tribes to be reunited. There are some Rabbi’s who believe that Joseph/ Ephriam is hidden within christianity. At least for the most part. The Tanach does say, that as long as Ephriam goes after Idols, they are to be treated as Goyim. What of Torah observant Natzarim, who do not worship pagan Idolatry, and don’t hold to a trinity concept. Should they be allowed in the Synagogues?
I don't know of such Christians though I am no authority on Christianity at all. The fact a non Jew might be of Jews offspring does not change our attitude to him - he remains non Jewish to a full degree. For the shul - these people should be treated like any non Jew in a shul - we let them come in but they do not share our rituals, they can come as observers.