Why interrupt the memory items to accelerate? Pitch down, the PM need not be involved? OK, FDs give "false" info, but that;s also the case during an emergency turn.
As long as you can communicate well and concisely enough to bring the PM with you and not just start flying outside the box and leave the PM unsure if you are in control, loosing SA or have a cunning plan. Two things I would most definitely want to avoid is loss of control or a confused partner.
Never even thought of keeping a burning engine running during climb to 1500 ft... We are so SOP drilled these days. Makes me think though, I like the idea.
I like to think that I'm not blinkered and that I would have the confidence to manage and balance the risks on the hoof (as it were). However, we don't operate from an airfield where a simple engine failure at V1 would mean we were unable to maintain terrain separation, so I can't see any reason to keep adding fuel to a fire to retain thrust.
I've heard a few people say that the plane is certified with the idea that the engine can burn on the wing for a while. If the engine had catastrophic damage, I don't know that I'd want the engine being fed fuel. If it was a fire with thrust, (stable N1 and N2) then I'd wouldn't mind keeping it running to get above the terrain if Boeing certified it that way.
I hear this from time to time, but I think maybe it is confused with the average survivability time of 17 mins for in flight fires based on actual non-survivable incidents! The Boeing FCTM is quite clear "Indications of an engine fire, impending engine breakup or approaching or exceeding engine limits, should be dealt with as soon as possible." I read that as 'all bets are off'
Worth a read too:
SKYbrary - Reflections on the Decision to Ditch a Large Transport Aircraft. Especially this bit "when you consider the whole episode from the start of the fire to touching the water took only six minutes, but when you’re on fire — that’s a long time".