I managed to imply the Neumann conditions on stress only by first 
projecting the boundary values to the entire mixed space (affecting 
rotations), and then manually removing the constraints associated with 
rotations. The code is below if someone is interested.
VectorTools::project_boundary_values (dof_handler,
                                      stress_boundary_functions,
                                      QGauss<dim-1>(2),
                                      boundary_values_stress);

FEValuesExtractors::Scalar rotation(dim*dim + dim);
VectorTools::interpolate_boundary_values (dof_handler,
                                          0,
                                          ZeroFunction<dim>(dim*dim + dim + 
0.5*dim*(dim-1)),
                                          boundary_values_rotation,
                                          fe.component_mask(rotation));
        
for (std::map<types::global_dof_index,double>::iterator it_r=
boundary_values_rotation.begin();
     it_r!=boundary_values_rotation.end();
     ++it_r)
   boundary_values_stress.erase(it_r->first);

MatrixTools::apply_boundary_values (boundary_values_stress,
                                    system_matrix,
                                    solution,
                                    system_rhs);


I still wonder, though, if there is a nicer way to do it.

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