It's my belief that a panel can still carry shear loads after exceeding the compression buckling load, but only if collapse in compression is prevented. (E.g., the compression is applied by thermal forces, which is essentially an enforced displacement, or there is other structure which will prevent excessive axial deformation*.)
Post-compression buckling the elastic forces in the panel do not drop, and continue to increase for additional applied compression; however the compression stiffness becomes so low that small additional forces over the buckling load cause very large deflections which rapidly lead to material failure. However, a lack of compression stiffness does not markedly decrease shear stiffness. (Though any applied shear will hasten collapse in compression.)
I'm not aware of any classical references which address post-buckling under combined loads. (I'm pretty sure that Timoshenko and Southwell both present elastic behavior post-buckling only under uniaxial compression.) So, the only way to assess post-compression buckling capacity for shear is by large-deflection non-linear FEA (unless someone knows better). Because stiffness never goes negative the usual Newton methods can be used for NL convergence.
* The additional structure can be quite low stiffness and low strength.
