If you haven't tried spray basting yet, I urge you to give it a try. After I have the quilt sandwiched, if it is a little off kilter, I find I can shift the fabric (either the backing or the quilt top, which ever is needed) by smoothing it repeatedly in one direction with my hands. You have to do this purposefully; it's not likely to shift on its own.
My kitchen bar area is long and wide. So far I have been able to sandwich the quilts I make on it. But I have used the basement floor, too. If I have to use the floor again, I will borrow my DH's knee pads.
Don't stretch the bottom fabric, just smooth it out. It all has to ease and settle together, and stretching tightly will make the other layers wrinkle once the stretched fabric is released.