There is the theory, then there is the reality. Disclaimer - I am not familiar with the B777, only more “primitive” jets.
From countless simulator exercises, I have observed that if a fire is introduced at V1 with simultaneous loss of thrust (highly unlikely, I know, but possible) and pilots attempt to complete memory actions for the fire drill at 400 ft simultaneous with, or closely followed by, acceleration and clean up, it is almost certain that:
Some item in the fire drill will be overlooked, due to the distraction/interruption of the acceleration and clean up procedure, or:
Some item in the acceleration segment will be mishandled - typically flap speeds will be exceeded or final climb speed will not be established on schedule.
If the fire is NOT accompanied by a simultaneous loss of thrust (more probable) and pilots attempt to combine the memory actions with an early acceleration it gets rather exciting as the power is cut about half way through the third segment.
So, I advise crews to: deal with fire ASAP, as it is the thing trying to destroy the structure, but keep climbing until the memory actions have been completed, then accelerate and clean up. For less critical failures, accelerate, clean up and establish final climb at MCT then complete the appropriate failure drill.
400 ft acceleration to gain some payload seems ill advised. 1000ft is a safer choice and easier for pilots to compute when surprised by a failure. The performance analysis can be tweaked to whatever acceleration above the legal minimum the operator chooses, albeit in some cases with a small payload penalty. It is probably time the legislation reflected that. 400ft was established way back in time when large piston engines and early turbine engines had very low time limits on takeoff power compared with today’s engines.
As for five or ten minute engine limits, again nice in theory, and the performance analysis has to account for this. But, in practice, what would the consequences be if a time limit over-ran by a minute or so in the interests of getting the job done? An engine is unlikely to self destruct, though of course may require premature overhaul any time a limit is exceeded.
At tea and bickies with the Chief Pilot I would be asking whether the airline preferred an overhaul to a hull loss.