Kerr parametric oscillators can encode Kerr-cat qubits, whose biased-noise nature can reduce the hardware overhead required for fault-tolerant quantum computation. In a recently proposed $R_{ZZ}$-gate scheme for Kerr-cat qubits, a tunable coupler must be displaced while avoiding nonadiabatic excitation. To address this challenge, we propose and theoretically analyze a protocol that applies fast-forward scaling to the coupler displacement. The protocol suppresses nonadiabatic excitation of the coupler and enables high speed $R_{ZZ}$ gates.