The face-centered-cubic lattice is composed of edge-sharing tetrahedra, making it a leading candidate host for strongly frustrated magnetism, but relatively few face-centered frustrated materials have been investigated. In the hydroxide double perovskite CuSn(OH)6, magnetic frustration of the Cu2+ quantum spins is partially relieved by strong Jahn-Teller distortions. Nevertheless, the system shows no signs of long-range magnetic order down to 45 mK and instead exhibits broad thermodynamic anomalies in specific heat and magnetization, indicating short-range dynamical spin correlations — a behavior typical of quantum spin liquids. We propose that such an unusual robustness of the spin-liquid-like state is a combined effect of quantum fluctuations of the quantum spins S = 1/2 , residual frustration on the highly distorted face-centered Cu2+ sublattice, and correlated proton disorder. Similar to the disorder-induced spin-liquid mimicry in YbMgGaO4 and herbertsmithite, proton disorder destabilizes the long-range magnetic order by introducing randomness into the magnetic exchange interaction network. However, unlike the quenched substitutional disorder on the magnetic sublattice, which is difficult to control, proton disorder can in principle be tuned through pressure-driven proton ordering transitions. This opens up the prospect of tuning the degree of disorder in a magnetic system to better understand its influence on the magnetic ground state.