Quasi-zero-stiffness (QZS) vibration isolators are effective for low-frequency vibration isolation but can exhibit undesirable large responses near their low resonant frequencies. This study proposes and experimentally demonstrates a passive resonance-avoidance strategy based on bistable switching between a high-stiffness (HS) state and a QZS state using a spring-steel compliant beam structure. Finite element analysis optimizes the geometric parameters to realize an HS region, a negative-stiffness transition, and a QZS region along a single force-displacement path. After fabrication, quasi-static experiments confirm the coexistence of two operating states, with the effective stiffness decreasing by 96.6% from the HS state to the QZS state. A nonlinear dynamic model incorporates the experimentally measured restoring force and state-dependent damping to compute the steady-state response branches associated with the local stable states. Base-excitation experiments show that state switching occurs mainly near the resonance band of the initial state. A barrier-normalized energy index analysis further supports the interpretation that switching is promoted by resonance-induced energy accumulation rather than by excitation amplitude alone. Relative to the locally continued non-switched branches, the switched responses exhibit frequency-averaged transmissibility reductions of 50.6 dB for the HS-to-QZS transition and 8.3 dB for the QZS-to-HS transition. Frequency-varying excitation tests at a constant amplitude also demonstrate successive bidirectional switching. These results show that the proposed HS-QZS beam structure can passively move away from resonance-prone configurations, providing a mechanical route for resonance avoidance in nonlinear vibration isolation.