A breathing pyrochlore system is predicted to host a variety of quantum spin liquids. However, perturbations beyond nearest-neighbor Heisenberg interaction are an obstacle to identifying such exotic states. Here, we utilize a bond-alternating disorder to tune a magnetic ground state in the Cr-based breathing pyrochlore. By combining thermodynamic and magnetic resonance techniques, we provide experimental signatures of a spin-liquid-like state in LiGa1-xInxCr4O8 (x=0.2), namely, a nearly T2-dependent magnetic specific heat and a persistent spin dynamics by muon spin relaxation (μSR). Moreover, 7Li NMR, ZF-μSR, and ESR unveil the dichotomic nature of both temporal and thermal spin fluctuations: slowly fluctuating tetramer singlets at high temperatures and a fast fluctuating spin-liquid-like state at low temperatures. Our results suggest that a bond disorder in the breathing pyrochlore offers a new route to achieve an unexplored state of matter.