In the field of vibration control, bistable nonlinear energy sinks (BNES) represent a highly efficient vibration suppression structure where energy harvesting and vibration control performance depend on the optimization and design of nonlinear system parameters. This paper investigates parameter identification in systems containing bistable nonlinear energy sinks. Leveraging the properties of stable equilibrium points in bistable systems, we propose a reference point shifting method that integrates Gaussian kernel density peak estimation to enhance the time-domain nonlinear subspace approach. Numerical simulations and experimental validations were conducted to analyze the impacts of various noise levels on equilibrium point estimation accuracy and nonlinear parameter identification, as well as the sensitivity of parameter results to steady-state point estimation errors. The results demonstrate that under 20% noise intensity, the equilibrium point estimation error remains below 10%, while the nonlinear parameter estimation error is approximately 20%. This method effectively identifies cubic stiffness and Coulomb damping parameters in bistable nonlinear energy sink systems.