Carpet just isn't the best place to sandwich a quilt, the nature of the beast means that you won't get it to lay perfectly flat and STAY perfectly flat.
My last quilt (which was admittedly only a lap size) I did using the Sharon Schamber method of using two boards to wrap the bottom and top in. It was PERFECT and easy. This method is shown in her video for basting a quilt (which is my preferred method of sandwiching, and I use water soluble thread to baste), but you could probably use the same method for spray basting or pinning. I would still however recommend a flat smooth surface to do it on.
Using her method though, you don't need space for the whole quilt - just the width of the quilt and a little extra. A long kitchen counter top, dining table, or even tile/wood floor would work.