Interesting question. It seems to me that the liquid on the outside comes from the condensation of water vapor in the surrounding air. Upon condensation on the bottle this vapor gave up heat to the ice-filled bottle thus warming it. When it re-evaporates it takes that extra heat away so it would seem to be an even game.
At the same time blowing air increases the rate of convective heat transfer so I think blowing air over the bottle would make the ice melt faster.
Any water that condenses and then evaporates is a break-even. On the other hand, water which condenses and then is blown away in small liquid droplets will have a net warming effect. The forced convection will also cause the bottle to more quickly reach equilibrium with its surroundings, and inefficiencies in the fan motor will add heat to the environment.
So we have four effects at work, one of which breaks even and the other three of which would tend to warm the bottle quicker. So the bottle will definitely warm quicker.