I know that copper door handles self clean, but does adding a penny to a vase actually kill bacteria build-up in water to the point that it makes a difference ?

https://www.yankodesign.com/20...

Not sure, but I doubt a penny has enough copper in it any more to make any difference.
Edited to add:
Sorry I did not notice the location you are at but the same might apply there also.

IF it works , use an old penny.

Cash is being phased out in the UK.

The article states that a pre-1992 1p or 2p coin from The Royal Mint, or a pre-1982 penny from the U.S. Treasury, were made from copper, but are since fashioned from Zinc.

The above tool made from copper is lengthy enough to do the job, but will it ?

You could always try it to see if it makes a difference.