We study a finite Floquet chain in which a uniform nearest-neighbor hopping coexists with a periodically rotating, SU(2)-dictated spin-assisted hopping profile. The resulting coupling is spatially inhomogeneous—weakest at the chain boundaries and strongest in the bulk—and produces a frequency-dependent Floquet-Sambe bottleneck. In the closed system, the mean inverse participation ratio (MIPR) of the Floquet eigenstates exhibits a striking nonmonotonic dependence on the driving frequency ω: the states remain extended at both low and high frequencies, but become maximally localized at an intermediate frequency. We demonstrate that this localization maximum occurs at ωpeak ∼ µ−s = √2s, a scale controlled by the first boundary bottleneck. To connect these spectral properties to measurable transport, we construct an open-system Floquet-Sambe Green-function inverse participation ratio from the spatial density of the injected scattering state. This open-system diagnostic recovers the same nonmonotonic localization trend as its closed-system counterpart, with the peak shifted to higher frequencies by the static bandwidth and the lead self-energy. These findings establish the driven synthetic spin chain as a directly realizable, frequency-tunable platform for coherent information storage and retrieval, rooted in the interplay of Floquet-Sambe virtual channels, boundary-controlled localization, and frequency-selective transport in emerging multi-level superconducting circuit architectures.