DNA is not precise enough to differentiate between Hebrew tribes that far back, or even between any other closely related groups then.
There's quite a lot of DNA evidence connecting today's Jews to each other and to other Middle Eastern people, and none connecting peoples of the British Isles to either.
The best the British-Israel folks can come up with is that the original Middle Easterners were all displaced to the northwest and all those people there now (as well as today's Jews) came from elsewhere.
I can post links if you're really interested.
The Lemba people would be Levites by ancestry, and Levi was a son of Leah, whereas Manasseh was a grandson of Rachel. Presumably Leah and Rachel were full sisters (at least there doesn't seem to be any indication that they had different mothers) but that would only be important if they were examining the mitochondrial DNA.
Has DNA testing been done on the Falashas of Ethiopia? Not everyone believes that they are really Jewish by ancestry and that would be a possible way to resolve the question.
I didn't raise the issue of the British-Israelites and have no interest in that topic.