Highly charged ions provide an outstanding platform for investigating bound-state quantum electrodynamics (BS-QED) [1-3]. One notable example is the study of the Lamb shift in lithium-like uranium [4-5]. Additionally, experiments have recently achieved exceptional levels of accuracy [6-11]. Combined with theoretical calculation, these achievements have made it possible to obtain the most accurate determination of the electron mass to date [12,13]. Continued development of theoretical and experimental methods opens up new possibilities for independent extraction of nuclear parameters, determination of the fine-structure constant and studies of physics beyond the Standard Model [14-16]. This progress also paves the way for the study of QED effects beyond the Furry picture in the strong coupling regime [16-19].

In this work, we investigate correlation effects within the Breit approximation using both the Coulomb and several screening potentials. We analyze the convergence of perturbation series for energies depending on the choice of the zeroth-order potential. The calculations are performed for helium- and lithium-like ions in both the ground and excited states. We employ the dual kinetic balance method [20] with a finite B-spline basis and the recursive formulation of perturbation theory [21], which allows perturbative terms of arbitrary order to be generated without explicit diagrammatic treatment. The results obtained will allow us to determine the limits of the applicability of perturbation theory to various states and to assess the effectiveness of using certain shielding potentials. This will ensure the correct application of the recursive perturbation theory apparatus, not only for calculating energies but also for determining the g factor and HFS.