Amplitude models are applied to studies of resonance structure in ${D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ and ${D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays using $pp$ collision data corresponding to an integrated luminosity of $3.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected by the LHCb experiment. Relative magnitude and phase information is determined, and coherence factors and related observables are computed for both the whole phase space and a restricted region of $100\text{ }\text{ }\mathrm{MeV}/{c}^{2}$ around the ${K}^{*}(892{)}^{\ifmmode\pm\else\textpm\fi{}}$ resonance. Two formulations for the $K\ensuremath{\pi}\text{ }S$-wave are used, both of which give a good description of the data. The ratio of branching fractions $\mathcal{B}({D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})/\mathcal{B}({D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{K}^{\ensuremath{-}}{\mathrm{\ensuremath{\pi}}}^{+})$ is measured to be $0.655\ifmmode\pm\else\textpm\fi{}0.004(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.006(\mathrm{syst})$ over the full phase space and $0.370\ifmmode\pm\else\textpm\fi{}0.003(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.012(\mathrm{syst})$ in the restricted region. A search for $CP$ violation is performed using the amplitude models and no significant effect is found. Predictions from SU(3) flavor symmetry for ${K}^{*}(892)K$ amplitudes of different charges are compared with the amplitude model results.