using RO is not ideal if the tap water has good level of alkalinity (i.e. < 100 ppm, ideally 40-60 ppm is best), and if tap water isn't hard. It's good to have low level of alkalinity in water to keep pH in soil solution from dropping too much (microbe exudates play a huge role in the pH of the soil solution and rhizophere, as does exudates of roots, which are acidic and basic).
If you add lime to RO water you are adding alkalinity (which is the level of bicarbonates, carbonates, etc), of which Ca and Mg have the biggest affect. Thus it's best to simply call your water supplier (if you're on city/town water) and ask what for the level of alkalinity; if it's over 100 ppm then you can filter it through peat moss which will remove some bicarbonates and carbonates, thus lowering alkalinity.
Good luck with your test, it's nice to see ppl experimenting!
However, what I wrote above is the best and most simple solution considering how much time it can take for RO filters to filter water, and then you are adding back what the RO filter removed? Kinda seem like ur chasing your tail (no offense intended).