Agreed a trolley with built-in timing smarts would be the closest exit.
But this being a discussion forum, this does identify a new tack. That is, you insert the dual-timing module inside the trolley. There are some loose-ends to tie up but then you would not have to mess with cutting the track rails or what have you. One could detect/sense the mechanical bumping action to trigger the delay timing module. Or the original trigger method I showed using a magnet and reed-switch could be used; in this case the magnet would be placed in the track bed near each end and the reed-switch would be placed on the bottom of the trolley. Power would be continuously applied to the track. The timing module's relay would cut power to the motor for 10-20 seconds when needed. $5-10 per trolley so it could add up if you need to modify a large fleet...as opposed to putting the electronics on the track/layout side.
This method has the benefit of allowing trolley interior LED/lights to remain ON when stopped since there is continuous power on the track! The passengers will surely appreciate this.