Are shift-chains two-colorable?
This is not an answer.  What follows is a simple proof that the construction for k=3 is indeed a counterexample.  I think that the asker knows this proof, but I will post it anyway because the proof is nice and this might be useful when people consider the case of larger k.

It is easy to verify that it is a shift-chain.  Let’s show that it does not have Property B.

In fact, the subhypergraph {(123), (145), (245), (345), (346), (347), (357), (367), (467), (567), (568), (569), (789)} already fails to satisfy Property B.  To see this, suppose that this hypergraph has a 2-coloring and let ci be the color of the vertex i.  Look at three hyperedges (145), (245), (345).  If c4=c5, then all of 1, 2 and 3 must be the opposite color to c4, but this would give a monochromatic hyperedge (123).  Therefore, it must be the case that c4≠c5.  Similarly,


c3≠c4 by comparing the three hyperedges (345), (346), (347) and noticing a hyperedge (567).
c6≠c7 by comparing the three hyperedges (367), (467), (567) and noticing a hyperedge (345).
c5≠c6 by comparing the three hyperedges (567), (568), (569) and noticing a hyperedge (789).


Therefore, we have c3≠c4≠c5≠c6≠c7.  But this implies c3=c5=c7, making the hyperedge (357) monochromatic.  This contradicts the assumption of the 2-coloring.