There are a lot of detailed considerations when computing electronic transport, but in general the answer is "yes", provided you can calculate (or estimate) a model Hamiltonian. For example, a DFT calculation may be used to compute the parameters for a model tight-binding Hamiltonian (e.g. using a local basis set, or Wannier90) which can then be used for much larger transport calculations.
Many studies use the (linearised) Boltzmann Transport Equations (BTE), for example as implemented in BoltzTraP2 or BoltzWann. A more sophisticated approach is to solve the quantum transport problem directly from the tight-binding Hamiltonian, for example the Quantum Kite software of my colleague Aires Ferreira and coworkers.
Although a few thousand atoms would usually require a small computer cluster for DFT (depending on the basis set), the tight-binding Hamiltonian can be solved much more quickly. A common limiting factor is computer RAM, and you may need to invest in more RAM to perform large calculations.
These methods typically give you the electrical conductivity per unit of time; the actual conductivity will depend on how long it takes for conduction electrons to relax back down to the (non-conducting) valence state, which is called the "relaxation time". The main factor affecting this is usually the electron-phonon interaction, but calculating that is computationally intensive. The common method is to use DFT, but this will not be practical for your situation; even if you did have the computational resources for a DFT calculation, an $N$-atom system has $3N$ phonons, and you need one DFT calculation for each phonon!
It may be possible to estimate the relaxation time, for example by calculating it for a prototypical bulk system (which has far fewer atoms), or by using acoustic deformation potential theory. Even if you do not know the relaxation time, you can still investigate differences in conductivity per unit time, and if you assume that the relaxation time is similar across the various structures then this trend should be reflected in the actual conductivity.