Assuming the the Book of Genesis we have, and the shape we have it
in, was available in pre-monarchical times, a reader would naturally
associate the Qadesh of Gen. 20:1 with the place he or she read about
earlier in Gen. 14:7. He or she also learned of a Shur (toward Egypt)
from Gen. 16:7. So by the time Gen. 20:1 was reached, the two place-
names you wish to associate with Lebanon have already been mentioned
and they have no connection whatsoever with Lebanon in those passages
that mention them. Thus to argue that Lebanon would jump into the
mind at Gen. 20:1 begs belief. It does not matter that Moses' Qadesh
and Shur have not been positively identified. I'm not sure why feel
you should keep reminding us. The biblical data still requires that
we look for a Qadesh in south Israel near Edom's border and a Shur
near Egypt's border, even if we never locate them. This is why your
theory, in my opinion, absolutely unbelievable. I also see no reason
to agree with the strawman you've set up: Gerar is not stated as
being situated in the desert, it is simply the next place the
biblical author wants us to know Abraham sojourned after his stay in
the south country.
"And Abraham journeyed from there toward the land of the south, and
he dwelt between Qadesh and Shur; and he sojourned in Gerar" (Gen.
20:1, translation mine).