So test for copper and find out. Letting it run for a while is a great idea. I didn't go for RO/DI just because I thought it would be a good geek thing to do. I'm on a well, and when it got to the dead of last winter, I started having algae from hell. Plants were all dying back, and my nice planted aquarium that looked so awesome to start with was a disaster. My instinct was to do more water changes, make sure the water was clean, also vacuum out the crud.
So it got _worse_. Then I asked a guy who is admin on the Fishaholics FB group (group is pretty fluffy, this guy is good), and he said I should test the tap water. Tap water was running 40 ppm nitrates out of the tap! Great, I'm doing water changes with fertilizer. Oh, yay. Now I'm buying 5 gallon jugs, schlepping them to the Fred Meyer, down the stairs to the basement, and it is costing I think about $0.50/gallon. Inconvenient, it is a workout, and expensive. Got the Spectrapure on order pretty quickly, couldn't wait to set it up - carrying 120 pounds of water down the stairs every couple days was not fun.
If you have acid water, like I do, then copper in the house is a very bad idea - had to replace all the incoming pipes for the entire house not long after moving in. That was _not_ cheap.
There are ways to pull metals out of things - the process is chelation. That said, I have no idea how one would accomplish that in a running aquarium. But if you can find something that binds copper, and isn't toxic itself, give it a try. I think I'd buy one of those test kits like ShortyKiloGyrl got, and find out how much of a problem you really have.