It oesn't seem likely that the Lost Tribes are going to come into play in the period around 2200 BC, since this is even before Abraham went to Canaan. Weren't the Lost Tribes scattered around 800 BC to 600 BC by the Assyrians and the Chaldeans?
What this event tat the end of the Egyptian Old Kingdom might relate to is the rise of the foreign (possibly Semitic) Hyksos kings over Egypt, who are believed by many to have been ruling when Joseph became advisor to Pharoah.
Here's an interesting link to a Ten Tribes article that I'd bookmarked:
No, 2000 BC is way too early.
What I think confuses many authors and scholars is that there were LOTS of different Hebrews roaming around that part of the world at that time. Abraham was just one, but his huge line of descendents gets virtually all the attention. Then, when some other group appears of historic interest authors try to relate those people to Abraham and his offspring. Makes for lots of misteaks IMHO.
For Example, many/most of the books on the CELTS are full of nice pictures of what is identified as Celtic art, but much of the narrative content is way off base. Fortunately, as result of good archeology in very recent years the Celts are being re-examined and re-defined quite differently. It is literally changing history.