I can see no reason that you can't move the waxed cheeses to a different place. If the other area is colder than the cave, it usually just means that it will age a little slower than usual. As long as they are turned regularly as usual they will be fine.
I am also getting into the habit of putting my waxed cheeses in the fridge to age, as I have so little room in my esky, and it seems to be working fine.