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High-fidelity two-qubit gates at scale are a key requirement to realize the full promise of quantum computation and simulation. The advent and use of coupler elements to tunably control two-qubit interactions has improved operational fidelity in many-qubit systems by reducing parasitic coupling and frequency crowding issues. Nonetheless, two-qubit gate errors still limit the capability of near-term quantum applications. The reason, in part, is the existing framework for tunable couplers based on the dispersive approximation does not fully incorporate three-body multi-level dynamics, which is essential for addressing coherent leakage to the coupler and parasitic longitudinal ($ZZ$) interactions during two-qubit gates. Here, we present a systematic approach that goes beyond the dispersive approximation to exploit the engineered level structure of the coupler and optimize its control. Using this approach, we experimentally demonstrate CZ and $ZZ$-free iSWAP gates with two-qubit interaction fidelities of $99.76 \pm 0.07$% and $99.87 \pm 0.23$%, respectively, which are close to their $T_1$ limits.