One of the best and most thorough teachings against the "two-house" theology is Ephaimites: A Case of Stolen Identity by Baruch ben Daniel. While you're on Baruch's site, please feel free to peruse the plethora of Biblically correct articles - and don't forget to return to The Refiner's Fire when you're done!

Below are excerpts from Baruch's article:

The "Ephraimite" movement has taken root out of prejudice and jealousy, it is not only divisive but it is taking a Name and Identity from others which is not rightfully to be taken. If one identifies himself as a Christian, Messianic, Netzarim, Believer, a Jew, Follower, Disciple, Talmid, a Spiritual Israelite, or someone with a Jewish Heart, Heart of a Jew, or Hebrew heart, they do so by their Faith. Anyone can choose to be a Jew, a Christian, a Netzari, a Believer etc, but to call oneself "Ephraim" without evidence to prove it, is by material point of law, theft.

Those who claim to be "Ephraimites" are also claiming to be a portion above their brethren, although a very small percentage of people having a love for Yeshua, Torah, Israel, the Jewish People, Hebrew, Aramaic etc claim to have any blood connection to Israel. "Ephraimitism" is an unsanctified division within the Body of Mashiyach and causing divisions within friends, families, and congregations. Most individuals are interested to grow their Faith in Yeshua and Torah, instead of being sold "Ephraimite" theories and "Lost Tribe" propaganda at Holy Convocations. It is wrong to hold meetings in the name of "Two Houses" and spend the majority of efforts and discussion around this motif, there are much more vital and important matters at hand. Prophetic passages in the Tanakh and Ketuvim Netzarim direct us to grow in HaMashiyach and go out and make Talmidim, thus creating the bond of Unity, we are not to focus on political and genealogical theorem.

...The fact of the matter is that Ephraim was very disobedient to YHWH and as a Tribe they never had such great military exploits as other Tribes, neither did they become the largest in number, far from it. Scriptures indicate that Ephraim let Canaanites dwell in their territories, the Canaanite idolatry rapidly brought Ephraim down physically and spiritually. To teach that Ephraim, irregardless of their crass rebellion to YHWH became or will become a great people, is antithetical to Scripture.