You can do all sorts of things to water chemistry during a fishless cycle because you get rid of it all before getting fish
In this case, you need to raise your KH to stop the pH dropping and put inorganic carbon into the tank for the bacteria. Looking at Seachem Neutral Regulator, it says it precipitates calcium and magnesium to soften water and it removes chlorine. It doesn't say anything about KH.
If you have any baking soda (not baking powder) in the kitchen, you can use that during cycling. That's sodium hydrogen carbonate and it will increase both KH and pH. Try 1 tablespoon per 12 gallons - dissolve it in a container with a bit of tank water then pour into the tank. Still keep an eye on the pH and if necessary, add a second dose if the pH starts to drop again. (If there isn't any in the kitchen, it's in the home baking section of the supermarket)