We report a measurement of the cross section for $K^0_S$ pair production in single-tag two-photon collisions, $gamma^* gamma to K^0_S K^0_S$, for $Q^2$ up to 30 GeV$^2$, where $Q^2$ is the negative of the invariant mass squared of the tagged photon. The measurement covers the kinematic range 1.0 GeV < W < 2.6 GeV and $|cos theta^*| < 1.0$ for the total energy and kaon scattering angle, respectively, in the $gamma^* gamma$ center-of-mass system. These results are based on a data sample of 759 fb$^{-1}$ collected with the Belle detector at the KEKB asymmetric-energy $e^+ e^-$ collider. For the first time, the transition form factor of the $f_2(1525)$ meson is measured separately for the helicity-0, -1, and -2 components and also compared with theoretical calculations. Finally, the partial decay widths of the $chi_{c0}$ and $chi_{c2}$ mesons are measured as a function of $Q^2$.