The company that I work for has a number of engine out procedures that require that have airspeed restrictions during clean up in the 3rd segment. To do this the flight mode has to change from take-off to speed select and level change resulting in the aircraft climbing above the engine out acceleration altitude until the point where the speed restriction no longer applies and then accelerating again to finish the clean up.
I believe that this procedure is unacceptable and have expressed my concern but no action is being taken.
From my experience in this area I maintain that the design of an engine out procedure must ensure that the 3rd segment can be flown as a level segment with no interruption to the level acceleration until the aircraft is clean.
While it is not a routine profile, it does occur on occasion. The reason for the speed restriction should be for terrain, it should not be for airspace/ATC/noise etc. If you get part way through flap retraction and get to a limit speed that has to be maintained, and you convert from a level acceleration to a climb at a constant speed, you are still compliant with the rules. At the end of the TO thrust limit time, say 5 minutes or 10 minutes dependent on type and AFM limitation, you are ..."obligated"... to set MCT. At that point if you are still speed limited, then you are required to have the OEI gradient in that condition to maintain level flight and accelerate in due course to clean configuration to conclude the 3rd segment. As the follow on segment has a minimum climb gradient in the clean config, while it will usually not be limiting, you may be performance limited if your circumstances ended up in a high drag config and back at MCT. You can probably assess the extent of a problem by comparing various WAT conditions that would approximate MCT for your blender vs TO thrust.
In the real world, the toss up between complying with a wall clock limit vs avoiding terrain should be fairly simple to prioritise.