That is what I would do.
Live rock doesn't draw much if anything into it so even if there is phosphates in the water, it will be taken up by fish, algae and other life forms before it gets into rock.
Phosphates and nitrates are readily taken up by algae and aren't a big problem unless you have live corals, which you won't have while cycling tank.
Heavy metals can be an issue to invertebrates (corals, anemones, crabs, shrimp, starfish, sea urchins, etc. Copper will kill any of these inverts but there does not appear to be any in your tap water (I had a look on the Lawrenceburg water company's website). However, the website did have some bad stuff in the water and they do recommend filtering tap water, preferably with R/O units. They also have the following written on their main page, which is a concern and suggests the water company does not believe the water quality is as good as it should be, and believes the government guidelines are not up to scratch.
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Quoted from
https://www.ewg.org/tapwater/system.php?pws=KY0030239
EWG's drinking water quality report shows results of tests conducted by the water utility and provided to the Environmental Working Group by the Kentucky Department for Environmental Protection, as well as information from the U.S. EPA Enforcement and Compliance History database (ECHO). For the latest quarter assessed by the U.S. EPA (January 2019 - March 2019),
tap water provided by this water utility was in compliance with federal health-based drinking water standards.
- Legal does not necessarily equal safe. Getting a passing grade from the federal government does not mean the water meets the latest health guidelines.
- Legal limits for contaminants in tap water have not been updated in almost 20 years.
- The best way to ensure clean tap water is to keep pollution out of source water in the first place.
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To me, this sounds like a good water company having a go at the federal government about not updating pollution laws.
Regarding heavy metals, some can be removed with carbon, and all will come out of the tank with a huge water change.
I have used copper in tanks and simply done a couple of big water changes, then added snails and shrimp and they were fine. So for cycling an aquarium, your tap water is fine but try to get a reverse osmosis system when you can. And drink R/O water from then on because there are a couple of things in your tap water that no animal (including dogs, cats, horses and anything else) should be ingesting.