I agree. If one has a rod or string which one divide into classical pieces of matter, then, when the rod undergoes length contraction, so does each of the pieces of the rod.The string in Bell's scenario doesn't get shorter, so the contracted binding EM fields have to span the same distances. Hence the tension. To avoid the complications of QM don't go down to the atomic level, but instead consider the contracting links of a chain that is forced to keep a constant length.
It doesn't matter to the argument how small each of the pieces is.
It does matter to the argument that we consider the pieces to behave classically. It's unclear to me how one rigorously deals with the quantum aspects, but this argument can go in another forum such as the quantum forum. I would guess that there is some sort of classical limit one can take, but I've never seen a serious formal discussion of the issue. This doesn't mean that one may not exist, as I'm not too familiar with the appropriate literature, unfortunately.