John I dont think you even need rain water once you use the brita filtered water.
I also used a basic brita jug to filter the water and then fill my rad when I do rad fluid changes and also for my water injection.
You know how good I find the brita filter ? I have a electric kettle that I boil water at the office every morning and during the days for the past 2.5 years. I use a brita jug just the regular one to filter the water.
Not once have I ever cleaned the electric kettle and when I open the lid to fill the water there is no build up of any chemicals/ residue nothing.
Even the the coils are perfectly clean.