We systematically study the effect of the inclusion of the frequency-independent Breit interaction - treated perturbatively as well as variationally - on Dirac-Fock total energies and ⟨r^k⟩ expectation values for He- and Be-like ions using an algorithm recently described by Reiher and Hinze. Fully numerical, highly accurate solution methods are employed throughout for solving the Dirac-Fock-Coulomb-Breit (DFCB) equations. These methods allow us to investigate the change of the wavefunction upon inclusion of the Breit interaction in the self-consistent field procedure of an atomic structure program. The dependence of expectation values on different finite-nucleus model (FNM) potentials for the electron-nucleus attraction is also studied. It is shown that, in general, the choice of the FNM for the electron-nucleus potential hardly affects the difference between Dirac-Fock-Coulomb (DFC) and DFCB expectation values. In the case of pointlike nuclei one should treat the Breit interaction variationally instead of perturbatively. For energy eigenvalues, we find that the difference between the Breit interaction treated self-consistently or perturbatively is negligibly small - even for highly charged ions - if an extended nucleus model is used. The data given may serve as a reference for more approximate treatments involving, for example, basis-set approaches with limited basis-set size. In this extensive study of the SCF effect on the Breit interaction, highly accurate, variationally obtained DFCB total energies and ⟨r^k⟩ expectation values are given for different models of the electron-nucleus interaction.