Walk in Humi

What kind of temperature swings did you get from the heating of the cement above? If the temp moved more than a few degrees during the day you probably need to readdress the insulation before you worry about a chiller. I recommend a minimum of R-33 in the ceiling and R-20 in the walls. And don't forget the floor. A cement floor will need at least R-20 to prevent heat seepage. Keeping a room cool is not the big issue for wine - it's the temperature stability. Every time your room moves a degree or two in temperature the expansion and contraction of the contents of the bottle will force oxygen through the corks hastening the decline of your wine.

I'll also reiterate, if you are going to chill the room to store wine, you are definitely going to want a vapor barrier on both sides of the insulation. That's because condensation will form anywhere there is a significant temperature gradient and you don't want that occurring inside your wall.

The units from Koolspace and Breezeaire are fine units and if you have insulated correctly you won't need a split system to cool your small room. Mount a unit next to the door exhausting into the hallway and you'll be fine. It will make a little noise; the fans are audible, but it will sound like a large refrigerator not a Cessna.