Yes, there are testers. I had the API one till it expired. It comes in a pack with a KH tester. They work differently form other testers. You don't add x drops, wait 5 mins and compare to a chart. You add the drops one at a time, shaking after each addition, until the colour changes. The number of drops need to make the colour change = the hardness in german degrees.
Filtered water is not a good idea. If you mean a Brita jug or similar brand, I know from testing the water that it lowers both GH and KH (and it drops my KH to zero) and it drops pH off the bottom of the scale, that is somewhere below 6.0.
Those water softeners that you plumb into your water supply can cause more problems. The Brita jugs swap calcium ions for hydrogen ions, and that is why the pH drops as the higher the number of hydrogen ions, the lower the pH. Some types of water softener swap calcium for sodium, and it is not a good idea to use sodium rich water in fish tanks. I'm sure you aren't supposed to drink this water either which is why this type must have a bypass tap.
Does it say on the pack exactly what the process is for the filtration system is? Filtration does sound different from softener. But I do know that brita style units can be fitted to the mains. If it's that type I wouldn't use those either. If it is
this I am not sure what to make of it other than thinking
at the price
As a matter of interest, do you know why water that has a lot of calcium, and maybe some magnesium as well, is called hard water? It is because it is hard to get soap to lather. Swapping the calcium for sodium makes it easier for soap to lather. I wonder why they used the word 'soft' for the opposite to hard rather than 'easy'. Maybe the term 'easy water' just didn't sound right