Consider this modified version of the Bell-Dewan-Beran thought experiment: Two spaceships are attached by a rope. They begin accelerating, with exactly equal accelerations, thereby maintaining a constant distance between them, as seen from the original rest frame.* Now what about the rope? It is moving too, hence it should contract, but since is stretched between the two ships, and the ships maintain their separation, it cannot contract. Eventually it will break, and people argue over this to this day, but this isn't what we want to focus on. Instead let's say the ships stop their acceleration before the rope breaks. Now the ships and the rope are moving at constant velocity, but the rope is still uncontracted. What happens next?
What happens next is we get to observe (mentally!) the contraction process happen in a very pure form. The rope will contract to its expected length, because only at that length are its atomic bonds free of strain. At the uncontracted length the bonds are highly strained and the rope is exerting a contractive force at each end, i.e., on the ships. It will pull the ships together until it shrinks to the expected contracted length. Then the contractive force ceases and the rope becomes slack (but of course the ships will continue moving closer, and will in fact collide).
The contractive force comes about because the fields between protons and electrons are deformed in the way originally calculated by Heaviside and noted by FitzGerald. It is quite real and capable of transferring momentum to other objects - namely the spaceships in this case.
So what about the idea that length contraction is purely a perspectival effect, not entailing any stress or strain? Well, this is true ONLY of the initial and final inertial states. These states are connected by the Lorentz symmetry, and if one is stress-free, then the other is also stress-free. But the Lorentz symmetry tells you nothing about how a system goes from initial to final state. It is, after all, only one of the symmetries of nature - not a complete description of it. Every effect predicted by relativity happens through some particular dynamical process and it is worth understanding the major sorts of mechanisms at work. I tried to compile them here .
* In fact the distance between the ships will not remain exactly constant because they are contracting as they accelerate; however, this doesn't change the point illustrated.