Crist,
I was writing my previous post as you were posting.
I think you are going to find a larger prop isn't going to use a lot more energy, simply because you are going to dial it in for your 5.1 s laps (at first at least). The larger diameter will give you more thrust, so you will need to turn down the rpm. My
guess is that the larger prop turning slower will be more efficient, and as such, your power draw will drop in level flight. Now I also think that the larger diameter will give you more thrust as the airspeed drops in the climb (which minimizes the loss of airspeed--which is good of course), so here you may pull more power than the smaller prop. So in the end, the total energy needed will the sum of the two effects.
I guess if I were you, and wanted to use the 2700 mAHr from the battery, I would up the diameter and lower the pitch--stay at 4" for example with your 13" prop. This should help lower the prop efficiency in level flight, but increase the vertical thrust in the climb (and braking in the dives). To get to 2700 might really require a low pitch (and high rpm). The high rpm should help burn power.
I think every plane needs a set amount of watts to fly, and to use much more you have to lose it in the prop. Of course if you gain in the verticals, it is a good tradeoff. The question is whether there is a point where the returns aren't worth it. I am sure there is a point, but not sure where it is at. Also I am not sure if we are already nearly there or not.