There seems to be a little bit of confusion creeping in here and I hope that I don't add to it!
IIRC the B737 manuals use the 400' height as the trigger to confirm the roll mode (HDG select) and to confirm the nature of the problem. If the problem is a problem that requires a recall drill then 400' is the earliest height that you should initiate any drills provided the aircraft is under control. The acceleration height remains at 800'/1000' AGL, or as specified by your take-off analysis procedure.
If at the 400' height the problem is confirmed as say an engine rundown then the drill is not a recall drill and can be left to a later stage.
Sudden winds without knowing the background as to why your operation has begun advocating this change in your procedure, I can only assume that like a couple of operators that I know of, they have decided that it is a reasonable compromise between operating the aeroplane safely and carrying out necessary non-normal drills. 400' is awfully close to the ground it is better to ensure the aeroplane is under control before any crew member begins carrying out checks.
A couple of points to consider. The Boeing manuals tell you that there are two instances that require immediate corrective action. The GPWS telling you you are about to hit the ground, and a stall. In almost every other situation you do have time.
In the case of an engine fire, severe damage, or seperation, just stop and think about these for a moment. Yes they are all recall drills, but even if the engine is on fire the chances are it is still producing plenty of thrust, so use it. A couple more seconds to let it get you up to the 1000' could be valuable and is unlikely to do much more damage to the engine itself. Also remember engines are supposed to be hot, the whole engine and pylon is designed to withstand heat, and the flames (if there are any) are being blown clear of the fuselage and tailplane. Over the years I have heard a number of stories of engines burning away on the wing until they fall off, and the aeroplane continued on safely enough. If you delay the drill for severe damage so what. The thing is already damaged, another few seconds will not make much difference, use those seconds to get to a safe altitude. Seperation, well the engine has fallen off anyway, youre just going to be going through the tidying up process, and without the weight of the engine it should be climbing well enough.