Cheapest way to heat and cool a converted garage?
If your current AC is not adequate to extend to the garage area, then a ductless mini-split wall mounted AC unit is probably ideal for your situation. Once installed, you'll likely exceed the $1500 budget, but I think it's the best of the options. You wall mount half of the unit inside the home and run the cooling lines through the wall and out to an exterior coil. They also have a condensation drain line that you run outside to help with the humidity.