Yes,the furr down can be done if as you suspect,their isn't enough space to add insulation. Foaming the cavity is another option. You shouldn't have to remove more than a couple of "peep holes" in sheetrock to check for insulation. The air might feel warmer or cooler because of volicity of air out of vent.
Before doing anything,you should aquire an accruate fast acting digital thermometer to test air temps. Record temp of air in supply plemum ( after air has passed over evaporator coil) or supply duct before duct enters floor. Subtract that temp from temp at supply grill(s). A difference of more than 0.75 degree on a day when garage is 85 to 100 degree suggest farther investigation is needed.I suggest you read some temps and tell us what you find.

I think that Klem is right. Although not stated specifically, the ducts should be inside the home envelope if you can do it. If you can, insulating the ducts themselves is not important. That means inside the insulation and vapor barrier.

You want to be sure that you do not have two vapor barriers. The position of the vapor barriers depends on your climate.

Pay attention to the walls as you fur down. You don't want to leave an area of uninsulated wall continuous with the "inside".

This post was edited by ionized on Wed, Oct 2, 13 at 19:26

You need to find out more about duct insulation and the location of the vapor barrier. Its always hard to keep heat even in an above-garage room, and it will be worse if they didn't the room and duct insulation correctly.

Cutting out and repairing a larger piece of sheetrock (say 12" x 12") is not much harder than a small inspection hole, especially if its in a garage where you may not require a perfect finish. If you cut carefully and keep the part you cut out, you can use metal drywall clips to put it back in the hole very solidly. Then use drywall tape and compound around the edges to make the repair. It takes a little practice to get the seams smooth, but just start out with the minimum amount of compound for the first layer and work up from there.

Bruce