Will increased precipitation in Antarctica prevent sea level rise?
There is some scope for continuing debate because quantifying the various components of the ice/snow/water balance are fraught with difficulty, and many of the estimates have error bounds which approach the magnitude that is being measured. However, a good best estimate, subject to continuing research, is given in:Is Antarctica losing or gaining ice? 
This cites the following two graphs:

 

Clearly there are important regional differences, with East Antarctica gaining slightly, but not enough to compensate for the ice loss in west Antarctica. 

It is true that the amount of fresh snow in the Antarctic is increasing, but it still doesn't amount to much. Antarctica is still the driest continent, by far, and much more so than Australia. The controlling factor is not the amount of snow on the land, but the fate of the sea ice around the perimeter. Notwithstanding climate deniers, there is abundant evidence that the oceans' sea surface temperatures are warming, including the Southern Ocean. This reduces the seasonal extent and thickness of sea ice, which lowers the oceanic albedo, and increases near-surface heat absorbtion. It is a well-documented feedback process. Overall the warmer ocean around Antarctic destabilizes the shelf ice, thereby unlocking the glacial ice which would have been held back. So we have two opposing non-linear processes, mobilization of coastal land ice (mass loss), vs increased snowfall (mass gain). At present, and for the forseable future, the mass-loss is winning. I'm guessing this will continue to be the case because the negative feedback is very powerfull. I don't think there are any reliable forward projections about how precipitation in the Antarctic will evolve. 

(Later) In view of the comments, maybe it would be helpful to visualize the daily precipitable moisture around the world. This is available at El Dorado Weather.