Ok, you are really making this more complicated than needed. The Oberth effect says that if you burn a given amount of rocket fuel at a low speed then you gain less KE than if you burn that same amount of fuel at a high speed.Trying to simplify the problem as much as possible. ...
So have two “pinewood derby” vehicles both on a straight slope, both with identical rocket engines. One fires the engine at the top of the track and the other fires the engine in the middle. Determine the final KE of each at the end.
Ignore trying to match the starting and ending times, the Oberth effect says nothing about times. Don’t have different paths, the Oberth effect says nothing about paths.