How to measure bacterial content
There is evidence that copper pots have antibacterial effects when used for storing drinking water (Sudha, et al. 2012), so the copper coins will probably have a similar effect.

The easiest way to test this at home would probably be to buy a test kit from an online vendor - these are relatively inexpensive.

Some Googling produced this website [appslabs.com.au], which has some tests for common types of bacteria. There are undoubtedly others too, if these don't sound suitable