Use pumps. They won't actually do anything but since there's no air circulation model anyway we'll pretend they do.
Build a ventilation system of 1-tile wide tunnels on the layer(s) above your active floors to act as your ceiling a/c ducts. Dig a channel above each room and install a floor grate to act as the a/c vent. install floor grates on the floor of each room that leads to another network of ducts underneath the floors.
Install a couple pumps at the air entrance to pressurize the air to get it moving and occasionally add additional pumps along the network of ducts to ensure that the air pressure stays high enough to keep circulating.
During the summer activate the above-room ventilation system that pulls air from the caverns. (Build a duct that goes to the caverns and put a grate over it. It'll keep flyers from entering it. And let's make the assumption it can also act as a filter to clean the air before you suck it in.) During the winter, enable the system of ducts that passes through your forge to warm the air. Make sure your a/c is connected to the ceiling vents and your heating system is connected to the floor vents. The fresh air will come out of the appropriate vent and air pressure will force it through the opposite vent where you can then reverse the pumps to pump the stale air away.
It won't accomplish a damn thing other than cosmetic, but that means it absolutely qualifies it as a megaproject. Do it! For science!
edit: Even better, forget the caverns for cool air. Instead do this.
Build a pump that pulls water from a local river and channels it into a line of copper pipes. Run the pipes down the middle of your main a/c intake duct which pulls in outside air instead of the hassle of going down to the caverns. Then the copper piping u-turns and discharges back into the river. As the cool river water flows through the main duct it will cool the air. Since you're taking it from the river, which is a constantly refreshing source, you will always be pulling in cool water. Once it's done its heat exchange job by cooling the air in the ducts (causing the water in turn to warm up) discharge it back into the river to be washed downstream.
Same principle as a real-world air conditioner, except instead of using coolant and a compressor you're using pumps and constantly refreshing cold river water.
If you want, run the main a/c duct through your forge area to heat the air first, then as you run it past the coolant pipes (of which you will have to make more to compensate for the extra heating from the forges) the warm air will more readily lose the humidity to the cold water pipes, conditioning and drying the air as it is pumped in.