  The conditional displacement (CD) gate between an oscillator and a
discrete-variable ancilla plays a key role in quantum information processing
tasks, such as enabling universal control of the oscillator and longitudinal
readout of the qubit. However, the gate is unprotected against the propagation
of ancilla decay errors and hence not fault-tolerant. Here, we propose a CD
gate scheme with fluxonium as the ancilla, which has been experimentally
demonstrated to have a large noise bias and millisecond-level lifetimes. The
proposed gate is applied cross-resonantly by modulating the external flux of
the fluxonium at the frequency of the target oscillator, which requires minimal
hardware overhead and does not increase sensitivity to decoherence mechanisms
like dephasing. We further provide a perturbative description of the gate
mechanism and identify the error budget. Additionally, we develop an
approximate procedure for choosing device and gate parameters that optimizes
gate performance. Following the procedure for multiple sets of fluxonium
parameters from the literature, we numerically demonstrate CD gates with
unitary fidelity exceeding 99.9% and gate times of hundreds of nanoseconds.
