Building large-scale quantum computers requires an interqubit-coupling scheme with a high on-off ratio to avoid unwanted crosstalk coming from residual coupling and to enable fast multi-qubit operations. We propose a (ZZ)-coupling scheme for two Kerr-cat qubits with a frequency-tunable coupler. By making four relevant states of the two Kerr-cat qubits quadruply degenerate, we can switch off the (ZZ) coupling. By partially lifting the level degeneracy, we can switch it on. We theoretically show that an experimentally feasible circuit model suppresses the residual (ZZ) coupling. Moreover, our circuit can realize (R_{ZZ}(-\pi/2))-gate fidelity higher than (99.9%) within (18) ns when decoherence is ignored. Our model includes the first-order terms in expansion beyond the rotating-wave approximation.