The conservative and dissipative dynamics of a 2DOF, system composed of a grounded linear oscillator coupled to a lightweight mass by means of both strongly nonlinear and linear negative stiffnesses is investigated. Numerical studies are presented aiming to assess the influence of this combined coupling on the transient dynamics. In particular, these studies are focused on passive nonlinear targeted energy transfer from the impulsively excited linear oscillator to the nonlinear bistable lightweight attachment. It is shown that the main feature of the proposed configuration is the ability of assuring broadband efficient energy transfer over a broad range of input energy. Due to the bistability of the attachment, such favorable behavior is triggered by different nonlinear dynamic mechanisms depending on the energy level. For high energy levels, strongly modulated oscillations occur, and the dynamics is governed by fundamental (1:1) and superharmonic (1:3) resonances; for low energy levels, chaotic cross-well oscillations of the nonlinear attachment as well as subharmonic resonances lead to strong energy exchanges between the two oscillators. The results reported in this work indicate that properly designed attachments of this type can be efficient absorbers and dissipators of impulsively induced vibration energy.