Weighing all of those on a balance scale will be very tedious. I would find a digital scale that has one more digit of accuracy than you need, that way it should be accurate at the resolution that you need.
Make yourself a set of test weights out of brass that are in the weight range of what you will be measuring. Mark the weight on them using a method that will not add or remove anything from them. (A marker adds ink and engraving removes metal. If you want nice engraved marking make the weights slightly heavy, engrave the marking then continue to remove material until you hit the weight that you marked.) Once you know the weight of the test weights you can always toss them on every once in a while to check your scale for drift in the range that you are measuring. Some scales will drift more around the high and low limits of their range, middle of range is the most accurate.
You do not necessarily need a scale that reads out in grains. You are just sorting by weight so any unit of measure could be used.
I do not have a scale that has precision to what you are looking for so I can not make a recommendation. All of my weighing is to see if part A or part B weighs more, I do not care what the actual weights are. All I need is repeatability.