We study the processes γγ→K+K-η and γγ→K+K-π0 using a data sample of 519  fb-1 recorded with the BABAR detector operating at the SLAC PEP-II asymmetric-energy e+e- collider at center-of-mass energies at and near the ϒ(nS) (n=2,3,4) resonances. We observe ηc→K+K-π0 and ηc→K+K-η decays, measure their relative branching fraction, and perform a Dalitz plot analysis for each decay. We observe the K0*(1430)→Kη decay and measure its branching fraction relative to the Kπ decay mode to be R(K0*(1430))=B(K0*(1430)→Kη)B(K0*(1430)→Kπ)=0.092±0.025-0.025+0.010. The ηc→K+K-η and K0*(1430)→Kη results correspond to the first observations of these channels. The data also show evidence for ηc(2S)→K+K-π0 and first evidence for ηc(2S)→K+K-η.