Long coherence times, large anharmonicity, and robust charge-noise insensitivity render fluxonium qubits an interesting alternative to transmons. Recent experiments have demonstrated record coherence times for low-frequency fluxonium qubits. Here, we propose a galvanic coupling scheme with flux-Tunable XX coupling. To implement a high-fidelity entangling $\sqrt{iSWAP}$ gate, we modulate the strength of this coupling and devise variable-Time identity gates to synchronize required single-qubit operations. Both types of gates are implemented using strong ac flux drives, lasting for only a few drive periods. We employ a theoretical framework capable of capturing qubit dynamics beyond the rotating-wave approximation as required for such strong drives. We predict an open-system fidelity of F > 0.999 for the $\sqrt{iSWAP}$ gate under realistic conditions.