What he said.
The reason Estes motors work for staging is the combustion chamber is inside the end burning BP. When it has burned close to the top of the grain, the grain fails and the internal motor pressure is vented through the ruptured top of the grain. This throws lots of chunks of burning BP forward and since it only takes one little piece to hit the BP near the nozzle of the upper stage motor to light it, that works quite well and is relatively dependable.
APCP motors have a central core and burn from the center of the motor to the sides. That central core extends all the way to the top of the motor. To light an APCP motor, you have to light it at the top of the grains, not the bottom near the nozzle. So not only can you not light an APCP motor the way you do a BP motor, it doesn't have the open top and bursting combustion chamber like an end burning BP motor does.
The reason for the 4 second minimum ejection delay on APCP motors is to prevent an ejection event while the motor is burning, you must have enough delay material in place to be structurally sound enough to handle the internal motor pressure. You don't want the motor pressure to burst the delay grain and ignite the ejection charge while the motor is still burning. When that internal pressure drops off at motor burnout, the remaining delay material that is thick enough to hold in the burning pressure takes 4 seconds or so to burn through at low pressure.
So, as Zeus-cat said, you must electronically ignite an upper stage AP motor.