I have heard about tap water that drops in pH when it sets awhile, but I still don't know what the water company is adding that dissipates as the water sets. I do know that all water companies treat their water to keep the pH above 7 at all times, so it doesn't corrode the water plumbing, whether copper or ancient lead piping. Have you looked at the water quality report for your water? Look for high sodium, for example, which might mean they introduce sodium in softening it. Look for an explanation of what they use to raise the pH.
You are buying RO water now, presumably storing it until you add it to the tank. So, I suspect it would be less work and bother to just fill a big container with tap water, let it sit open to the air, perhaps with an airstone in it to help it along, and don't use it for a couple of days or more. You can use a small pump to transfer the water to the aquarium.
Another thing to consider is doing smaller water changes, and using tap water directly. If you now do 50% changes weekly, switch to 20% changes every other day. That way you are altering the tank water properties a lot less with the water changes. And, don't forget to use Prime or equivalent de-chlorinator with each change.