This　paper　investigates　the　dynamics　of　a　novel　bistable　2-DOF　coupled　oscillator　with　nonlinear　damping.　The　system　is　based　on　a　conventional　bistable　nonlinear　energy　sink　(NES),　with　nonlinear　damping　introduced　to　enhance　the　vibration　absorption　efficiency　and　reduce　the　threshold　for　a　strongly　modulated　response　(SMR).　Initially,　the　slow　invariant　manifold　(SIM)　of　the　system　is　derived　using　complexification-averaging　and　multiple　scales　methods.　The　characteristics　of　saddle-node　(SN)　bifurcation　and　Hopf　bifurcation　at　the　periodic　fixed　point　are　then　analyzed,　identifying　the　ranges　of　stiffness,　damping,　and　other　parameters　that　influence　the　SMR.　Subsequently,　the　SIM　is　examined　in　detail　across　different　timescales,　and　the　amplitude　thresholds　of　external　excitation　that　trigger　SMR　are　derived.　Finally,　the　system＇s　maximum　amplitude　is　optimized　to　reduce　it　while　ensuring　the　generation　of　SMR.　A　numerical　analysis　of　the　energy　spectrum　near　the　resonance　frequency　is　also　conducted　to　compare　the　vibration　suppression　efficiency　between　the　conventional　NES　and　the　bistable　NES　with　nonlinear　damping.　The　results　demonstrate　that　the　bistable　NES　with　nonlinear　damping　effectively　lowers　the　external　excitation　threshold　for　SMR,　offering　a　wider　threshold　range　and　higher　vibration　suppression　efficiency.　Key　parameters　such　as　the　mass　ratio,　linear　and　nonlinear　stiffness,　and　nonlinear　damping　significantly　influence　the　occurrence　of　SMR.