My thoughts:
Firstly
You have to find a way to attch the menntos in the bottle otherwise it will fly out with the first interaction with coke.
The hard part seems to be to put the coke inside the bottle with water without mixing and without touching the mentos too early. An idea which I don't know if it works is that you take a very thin plastic bag which you fill with coke and put inside the bottle, at mthe bottom. Then you take a metal wire (very thin to not increase the weight) and make two holes in the bottle just above the thin plastic bag and put the wire through the holes so that if you turn the bottle around (ie in starting position) the bag lies on the wire. Now you put some tape on the hole just to nnot let escape air from there (stability).
Now you take one ore more mentes and bite them or cut them just so that they have a sharp corner. Now you take some glue put it on the sharpened mentos and with some chopsticks you try to stick it/them to the wire.
Now the theory is that the thrust of the water going out will push the coke bag against the sharpened mentos and if everything goes well the bag breaks and you have your seeked second fuel system coming into action.
Now the problem I see is that I don't know if the reaction is quick enough, in the sense is it enough that the cokes touches a mentos and it starts right away?
Another idea is you put the mentos and the coke already mixed in a bag (close it quick enough

) and put something else on the wire so that it will break during the "water thrust".
A general problem I see is: does the coke mentos reaction also happen when there is enough volume? I mean I only tried with a full bottle never with an 3/4 empty one.
Last suggestion: use out of date coke (it still works I tried only with 2 years out of date coke) because:
1) You don't waste something still drinkable
2) You have a better conscience
3) You might find it at a cheaper price or for free so you can make many tries.