“… is any chance at all of it flying with two engines gone after V1,”
Yes, providing the aircraft is flown at or above Vmca2.
In those aircraft which are allowed to dispatch with an engine inoperative (ferry flight) this is the basis of the take off calculations. The takeoff speeds are chosen so that V2 >= Vmca2.
There may be specific handling procedures when on the runway to accelerate the three good engines to mitigate the effects of Vmcg1, i.e. two engine roll, the third engine accelerated at higher airspeed. Calculations also take into account take off / climb weight, which generally is not a problem for ferrying.
In some aircraft types / situations there may be a small period of risk if an engine fails after V1 but before V2 (Vmca2). At this point probabilities come into the equation e.g. is the failure an asymmetric or symmetric engine, will it occur between V1 and V2. IIRC there are differing certification standards in this area, i.e. some authorities allow the risk, others require V1=V2>Vmca2.
The requirements and resultant probabilities in certification are the bases of not considering a four engine take off with two failing at a critical stage of take off. This is not to say it will not or has not happened, but it is sufficiently remote not to worry about (train) at a critical stage of flight. Where such failures have occurred some crews have been able to mitigate the assumptions of certification e.g. IIRC at BAe146 lost 2.5 engines shortly after rotate (TNT, Rome, Geese??).