This sounds interesting and I wouldn't mind trying it for my next batch. Does anyone have the specs on the paper they use? The nearest Kinkos is a half hour drive each way, but there is a UPS store in my town.
I use cheap cheap office depot paper. If you get both sides of the paper wet with milk, it should not curl. If you only do one side of the paper, it will curl. So either dunk the paper completely under the milk, or brush (or spray) the front of of the label after you apply it.
And keep an eye out for cats until it dries.