Ok, so to be clear, this testing is from a company that is going to simply parrot the standard pool industry recommend levels without regard to much logic. Yes, your saturation index is probably very negative, but you have a vinyl liner pool and so there is no need to have water that is oversaturated with calcium. TFP typically recommends a minimum of 50-100ppm CALCIUM hardness (not general or total hardness). At 25ppm your water is low in calcium but that is only really going to manifest itself, possibly, as water foaminess. If you raised your CH up to 100ppm, that would be totally fine. So, not to get too technical, but TFP has had this discussion many times in the past and saturation indices (whether you use LSI or CSI or some other index) ONLY give you an idea of how saturated or unsaturated calcium is in the water, that has NOTHING directly to do with metal corrosion. The assertion that because your water is not fully saturated with calcium that it will somehow look for metals to corrode is nonsense. Saturation indices ONLY apply to calcium, not metal corrosion. Metal corrosion, for the most part, is dominated by pH, TDS/conductivity and whether certain ions, like chloride and sulfates, are present. Your water pH seems a little low and if your tap water is normally that low, you can certainly raise pH a bit. But I think you have been keeping your pH on the low side because of the stains, so that explains your low pH not what your normal pH levels would be. I honestly would ignore most of their recommendations.
The copper metal came into your water form the algaecide that they told you to use. So, at nearly 0.5ppm, it is high enough to cause stains. The only way to get rid of copper is to exchange pool water with fresh, copper-free water. That can be done with draining and refills or perhaps draining before a big rainstorm and using rainwater to refill the pool.
If you don't have a pool heater and there's no metal in the pool, then there's really no issue with your saturation index.