You might want to consider, say, a Berkey filtration system. It isn't attached to the sink at all -- it works through gravity -- and it allows for superior filtration. However, there will still be mineral loss. There doesn't seem to be a way to avoid that, unless your filter includes a re-mineralization section. Instead, I'd use things like stock and mineral baths, etc., to help fill in the gaps.

I owned a Berkey filter with the carbon and fluoride filter and I was disappointed with it. It is a very expensive filter and they do not tell you the, but mold developed in my lower chamber where the filtered water was stored and it also appears that white dots are also on my carbon filters which may be mold also. I scrubbed and scrubbed the carbon filter with a wire scrubby but the white dots did not disappear. I contacted Berkey and they said that they never had this problem before. Also I tightened the fluoride filter too much that I broke the membrane and alumina particles were floating in my filtered water. I was told by Berkey that it may be toxic and that I should discard the water and order new fluoride filters. After discussing this with my husband, we decided not to get any more filters, but we cannot return the filter and get our money back, it has passed 30 days. So please save your money and get something else. We have learned an expensive lesson.

The most effective technologies for removing fluoride are distillation and reverse osmosis, which also remove everything else. Thing is, that everything else can be anything from beneficial minerals to pollutants. Our household water is rainwater off the roof, but we run our drinking water through reverse osmosis because it removes any mercury pollution the rain picked up from the atmosphere. We then use a mineral cartridge to put minerals back in. Personally, I like having complete control over what's in my drinking water.