You probably had a hard time finding an answer because it isn't straightforward. There are many factors which can influence the best approach, and the times.
For example, the temperature of the surroundings and the insulation of your container will have significant effects on the temperature over time. A highly insulated container will allow you to reach a lower minimum temperature, and cooler surroundings will do the same. Placing the bottle in the freezer will also lower the minimum possible temperature (even if you take it out of the freezer to do this experiment, a cold bottle is better than a warm one).
The initial temperature of the drink compared to the initial temperature of the ice are also very important.
To determine when it will be the coldest, you need to compare the amount of heat transferred from the ice to the drink, and from the surroundings to the drink. As long as the rate of heat transferred to the ice is greater than the rate of heat transferred from the surroundings to the drink, the system will be cooling. I would expect to find the lowest temperature near the point when the ice is melting. This is because melting ice requires the latent heat of fusion, and ice retains it's temperature during this process; so the ice can absorb heat while still remaining at the same temperature. This phase transition represents a fairly large portion of the cooling ability of ice.
The specifics of when this will happen can vary a lot depending on your setup though. Factors such as the geometry of the container and the geometry of the ice can be significant in determining the heat transfer between those objects and their surroundings (for example, surface area effects the rate of heat transfer, and geometry can effect convection currents).