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 compared with theoretical calculations. The $\gamma^* \gamma$ partial decay widths of the $\chi_{c0}$ and $\chi_{c2}$ charmonia are measured as a function of $Q^2$.