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We study the processes $\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}\ensuremath{\eta}$ and $\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ using a data sample of $519\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ recorded with the BABAR detector operating at the SLAC PEP-II asymmetric-energy ${e}^{+}{e}^{\ensuremath{-}}$ collider at center-of-mass energies at and near the $\mathit{\ensuremath{\Upsilon}}(nS)$ ($n=2,3,4$) resonances. We observe ${\ensuremath{\eta}}_{c}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ and ${\ensuremath{\eta}}_{c}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}\ensuremath{\eta}$ decays, measure their relative branching fraction, and perform a Dalitz plot analysis for each decay. We observe the ${K}_{0}^{*}(1430)\ensuremath{\rightarrow}K\ensuremath{\eta}$ decay and measure its branching fraction relative to the $K\ensuremath{\pi}$ decay mode to be $\mathcal{R}({K}_{0}^{*}(1430))=\frac{\mathcal{B}({K}_{0}^{*}(1430)\ensuremath{\rightarrow}K\ensuremath{\eta})}{\mathcal{B}({K}_{0}^{*}(1430)\ensuremath{\rightarrow}K\ensuremath{\pi})}=0.092\ifmmode\pm\else\textpm\fi{}0.02{5}_{\ensuremath{-}0.025}^{+0.010}$. The ${\ensuremath{\eta}}_{c}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}\ensuremath{\eta}$ and ${K}_{0}^{*}(1430)\ensuremath{\rightarrow}K\ensuremath{\eta}$ results correspond to the first observations of these channels. The data also show evidence for ${\ensuremath{\eta}}_{c}(2S)\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ and first evidence for ${\ensuremath{\eta}}_{c}(2S)\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}\ensuremath{\eta}$.