Wow... sounds like you're pretty set on using your tap water, so why ask about the levels to start with? You asked for opinions, no? Not that it'll make a difference, but there are two problems that I see...
Local water providers are only required by the feds to test their water once a year. Once. And those are the nifty results they send out to you and tell you over the phone. I'm not saying they don't test it more often - I'm sure they do. But the results that are for public consumption are those yearly tests that they have to submit. Just because the water is "perfect" when they test, doesn't mean it will be "perfect" the following week. When I started my tank, I tested my tap water - 0 nitrates. Within 6 months, for whatever reason, I was getting 5-10ppm nitrates in the tap.
The other problem is that there's a lot of pipe and miles between wherever your water provider tests, and your tap. Your pipes within your house could be leaching material into your water that won't show up in the water provider's test. For example... I use a Kold-Steril unit for one of my first stages of filtration. One of it's stages uses PolyFilters, which change color depending on what it is adsorbing. After a year use, the PolyFilters in my unit are a very pretty blue color. They turn blue when sucking up copper. According to my water department, I have no copper in my water. It's not hard to imagine that with the amount of copper piping in my house, that this is where it's coming from.
Obviously, you'll do whatever you choose. The filtration you describe is better than nothing - for sure. But those fridge filters or the Pur/Brita type filters really don't take out as much stuff as you'd think. I have 88ppm TDS (total dissolved solid) water coming out of my tap. After trickling it through a Brita, I still have a TDS reading of around 40-45ppm. Better than nothing, but not pure by any means.
No... your fish won't die from using tap water, even without any of the filtration you mention. In my opinion, in the long run, you will end up costing yourself more money buying tap water conditioner, and replacing the filtration cartridges in your fridge versus getting a cheapo RO/DI unit.