A measurement of \(CP\)-violating observables associated with the interference of \(B^0\to D^0 K^{*}(892)^0\) and \(B^0\to \bar{D}^0 K^*(892)^0\) decay amplitudes is performed in the \(D^0 \to K^{\mp}\pi^{\pm}(\pi^+\pi^-),\) \(D^0 \to \pi^+\pi^-(\pi^+\pi^-)\), and \(D^0\to K^+K^-\) final states using data collected by the LHCb experiment corresponding to an integrated luminosity of \(9\) \(\text{fb}^{-1}\). \(CP\)-violating observables related to the interference of \(B^0_s\to D^0 \bar{K}^*(892)^0\) and \(B_s^0\to \bar{D}^0 \bar{K}^*(892)^0\) are also measured, but no evidence for interference is found. The \(B^0\) observables are used to constrain the parameter space of the CKM angle \(\gamma\) and the hadronic parameters \(r_{B^0}^{DK^*}\) and \(\delta_{B^0}^{DK^*}\) with inputs from other measurements. In a combined analysis, these measurements allow for four solutions in the parameter space, only one of which is consistent with the world average.