Some thoughts ..
(a) twins are comparatively over powered to allow for the OEI case .. thrust is down by half but the resulting performance is down by something in the region of 80 percent
(b) for a near-V1 failure, V2 is important as a balance between distance and climb gradient when compared to what is scheduled in the AFM. If you try to climb much below V2, performance is down and other margins (Vs, Vmca, for instance) are reduced. If you try to accelerate to V2 + delta, you spend distance which might compromise runway numbers and obstacle clearance
(c) on the other hand, if the failure occurs shortly after takeoff, the AEO acceleration has put you at V2 + delta and you are better off hanging onto some of that delta in accordance with OEM guidance (typically V2 + 10-20)
(d) if you are WAT-limited, you don't have much any sort of spare performance and the available speed range will be fairly narrow ... (b) and (c) still apply
(e) obstacle scenario depends on location and balancing distance gone at failure against actual speed against required gradient .. all in all one does the sums beforehand and then sticks to the script
(f) what the box commands will depend on the certification and whatever programming is embodied