In terms of all the arguments that make up the peculiar Britam variety of British-Israelism, the thematic reasons (rather then historical and genealogical reasons) for associating America with Manasseh were always the most compelling to me.
I addressed this briefly in the American Antichrist post. I'll go over them again, probably in more detail.
The continental United States like the land allotted to Manasseh is divided between East and West by a major River (Mississippi/Jordan).
People have suggested a possible etymological connection between HaMakir (Makir/Machir was the first born son of Manasseh) and America. The only issue I have there is how the k/c and the r became switched around. It's been suggested the concept of Representative democracy is implied in the very name of Manasseh, as well as Machir. Those claims seem questionable to me however.
What's most interesting is how I felt long before I heard of British Israelism that George Washington had a lot in common with Gideon. Both were military leaders of successful rebellions, both were offered to be King but declined the Crown, and both become symbolically represented by a Tree.
Now I'm not a believer in British-Israelism. I think rather that the deported Northern Israelites went East rather then West. But that could potentially include them coming to the Americas and contributing to the Native American gene pool. I also think well before the deportation the Israelites had awareness of the Americas via their complex relationship with Tyre and the Phoenicians. I highly recommend the book America BC for documenting much evidence of the Phoenicians being among the Pre-Colombian travelers to the Americas.
Lots of intermingling between Native Americans and the Western colonists went on, really a majority of even "white" people in America today likely have some degree of Naive American ancestry. That does not take away that by looking White we still have White privilege.
In Genesis 48:19 Jacob Prophesied that a Great Nation would come out of Manasseh and that many nations would come out of Ephraim.
Making America Manasseh in the context of Western Nations as the Lost Tribes usually goes with making Ephraim the British Isles and the other Commonwealth nations, Queen Elizabeth II is in fact the Queen of 16 different nations. However I only see Edom and Dan when I look at England, (well also some Japheth clearly). The Lion as a symbol of theirs is used to justify connecting them to Judah/David, but in Deuteronomy 33 Dan was also called a Lion's Whelp, and the current British Royal family is really Danish, both the Queen and her consort descent from different children of King Christian IX. And before that the Angles and Saxons came from Denmark.
However if it's through Native Americans that the United States could possibly claim Manassehite heritage, then perhaps Mesoamerica/Latin America (Mexico, Central and South America, and some of the Caribbean Islands) is Ephraim. Mexico is to the south of the Western United States just like Ephraim was to Western Manasseh.
There seemed to be some shifts at different times in terms of what was Manasseh and what was Ephraim, Shechem and possibly Samaria seemed to be affiliated with both. That area was originally allotted to Ephraim in the time of Joshua, but in Gideon's time it seemed like the capital of Western Manasseh. But it seems to be Ephraim again by the time of Jeroboam.
Likewise the Southwestern United States once belonged to Mexico.
In the American Antichrist post I made a thing out of the Bald Eagle being associated with Samaria in Micah chapter 1. Well Mexico has an Eagle on it's Flag derived from Aztec legend. Aztlan is frequently speculated to have been in the American South West. It's interesting that the South West is where the LDS Church ultimately set up it's headquarters. Mormonism uses and twists for their own purposes the idea of the Native Americans being Israelites, in particular of Manasseh, yet at the same time identifies itself with Ephraim.
Shiloh and Bethel were both in what was pretty consistently Ephraim. Perhaps the analogue to Shiloh which was Israel's capital often during the Judges period is Mexico City. And Bethel's would be Teotihuacan. Or visa versa.
On the subject of Manasseh and the Jordan/Mississippi connection. Since I now support a north of the Dead Sea location for Sodom and Gomorrah, it might amuse some to know that puts them in relation to the Jordan about where New Orleans is in relation to the Mississippi.
Britam's desire to affiliate Ephriam with monarchical form of Government is also weak. They talk like all the Kings of the Northern Kingdom were of Ephriam or at least Joseph, but really only Jeroboam is confirmed, Bassha we know was Issachar and there is evidence Menahem came from Gad. Genesis 49 and Deuteronomy 33 gives Royal Promises only to Judah and Gad, and maybe Dan and Asher. Samuel was of Ephriam and voiced a strong anti-royalist sentiment.
None of this takes away the Edomite Heritage that the modern American Nations have via Europe/Rome. In the days of Gideon the Amalakites had a fortress in the Mountains of Ephriam. And a connection between Ephriam and Edom is implied is implied in Obadiah as well.
Another thing I think Britam might be right on is Moab and Ammon being Spain and Portugal, Moab is also affiliated with Eagle imagery in Jeremiah. Britam however wants to take from that expanding it to Latin America. But Moab and Ammon were among the nations that oppressed the Israelites during the Judges period. So perhaps Ephraim was under the yoke of Moab and Ammon for centuries, but then after the American Revolution (because Manasseh was the first born) Ephraim started throwing off the yoke of Moab and Ammon under leaders like Simon Bolivar.
One thing I want to make clear is my speculating on Northern Israelite heritage for any Asian or Native American peoples does not take away that other people were there first, with possible Genesis 10 ancestry independent of Shem much less Abraham. But after the deportation most of the northern Israelites lost their identity and were absorbed into and intermingled with various other peoples.
Update December 1st 2017:
First of all, it seems from my research since I made this, that it was chiefly and possibly just Shechem that was affiliated with both Ephraim and Manasseh. Shechem is also where Joseph was buried. Samaria being usually placed north of Shechem is most likely Manasseh territory. It seems the confusion is added to because the mountain range called the "Mountains of Ephraim" (Sometimes translated as if it were singular in English) extend into Manasseh's territory.
Given the elaborations that have since been made to my Lost Tribes views. I no longer think Native Americans are the only means by which Manasseh came to America. It was those carried away by Assyria that were taken East and wound up in the Mississippi region.
In a future post I'll be talking more on what I think about the fates of the Northern Israelites who weren't carried away by Assyria. Where I will explain that I think descendants of Western Manasseh (and to an extent other tribes) wound up in America via Irish&Scottish Immigrants (Welsh and Bretans as well) and African Slaves (and also Irish Slaves).
Deuteronomy 23:7-8 says that Edomites and Mizraimites who come to dwell among the Israelites should be let into the congregation after the third generation. By the American Revolution quite a few colonists had been here for more then three generations.
I've argued on this Blog that the Scandanavins, and other Germanic Tribes including the Anglo-Saxons and Dutch chiefly descend from Dan and Mizraim (Jeremiah foretold Mizraim winding up in the North). There is also likely a strong Danite element still in Greece.
I also still feel inclined towards identifying Edom with Rome, in-spite of that recent Daniel 7 post I made. Rome contributed to the population of the British Isles and other parts of Europe, but they would chiefly be Italians (and Corsicans). There were some Italians already here before the Revolution, they mostly came from northern Italy, the waves of Italian Immigration in the late 1800s and early 1900s were chiefly from southern Italy and Sicily.