First, what is the "cooperate-graduate" method? I am not familiar with this terminology.
Second, where does the 250' tolerance on acceleration altitude come from?
Although fully understanding the first part (long time modus operandi at some airlines), I also wonder just where this 250 foot tolerance comes from...
FAA inspectors generally presume that the pilot, once an engine fire is annunciated just after takeoff (and excluding severe damage/heavy vibs etc) would expect that engines thrust to be used until the end of the acceleration phase, before fire drills are commenced, whereas...most EU ops want the fire drill started at 400 feet, no later, period.
Having operated under both scenarios (for sim training) I follow the expected laid down company procedure, as this is far easier than trying to educate myopic checkers in the (correct) FAA procedure.
IE: Some of the EU trained guys are apparently of the (old) deHavilland Comet mindset (engines buried in the wing, not on pylons, as they are today), and want fire drills started at what I (and the FAA) consider, unacceptably low altitudes.
So, I would suggest for the original poster, do as your checker says, for much easier progression.