Even with the board insulation--I hope it's extruded polystyrene--you need mechanical dehumidification to keep humidity below 50%. Especially as you have carpet, which is partial to mould growth. (BTW, I never heard of carpet layers using water.) See Building Science.

The finished basement in my current home is insulated the same as yours, but with open cell in the rims. In the summer I pull out at least 20 pints a day from about 1650 sf.

Our basement is always at least 10% more humid than the house, sometimes more. The only time it hits 50% without a dehumidifier is in winter.

My understanding is that AC is inadequate to remove humidity from a basement because basements are typically cool and the AC can't run long enough to remove the humidity. We have a bone-dry basement and run the dehumidifier from May until early October. I'm surprised you even need AC in the basement, ours is only 60-65 degrees in the summer.