The coins minted by the US have always had very accurate composition. The problem is that to get an accuracy of 0.005 (i.e. 0.5%) in the silver content, you will have to measure the specific gravity to an accuracy of around 0.005 * 0.17 = 0.00085 or 0.085%, as the difference in specific gravities of copper and silver is only about 17%.
And then, because of the way you're measuring the coin weights in and out of water, to measure the specific gravity to an accuracy of 0.085% you will have to measure the weights to an accuracy of around 0.00085 x 0.10 = 0.000085 (since the weight in and out of the water differs by about 10%).
So to measure the silver composition as accurately as your smelter buddy, you're going to have to weigh them in and out of water with an accuracy of around 0.01%, i.e., you need to be measuring with 5 significant digits. In short, to get results that don't bounce all around you're going to have to invest in quite the scale.
Here's a table for silver / copper (gold/copper would be a lot easier):
5 digits: 0.5%
4 digits: 5%
3 digits: 50%
The above are worst case and are my back of the envelope guesses. You should be able to do better, depending on how lucky you are. (For example, if you have 3 digits accuracy you might measure 11.3 or 95.7. Then 0.1 is around 0.1% of 95.7 but closer to 1% of 11.3.)
As it turns out, this problem is one I've faced in my primary hobby, physics. This is why you need to give "error bars" on every measurement you make. And then, when you do the computations, you have to include the error bars with them.
I can type up an example showing how to do error bar calculations if you're interested. Doing calculations with error bars is a pain, so when I had to do this for hundreds of data items in one of my papers (on 100+ hadron masses), I wrote a java applet to take the data and do the calculations automatically for me.