Why not simply put the flat tire on the rack just like the spare was ??? Aren't they the same size, unless you have a doughnut?
I have Continental tires, 235/40 R 19 size. They are 27" diameter and 9.25 width. One flat tire with wheel will fit under the rack. When you have a flat, the procedure is to remove the rack with the mini spare ( dough nut) , unbolt the spare front the rack buy hand loosening the wing nut that holds the tire down via the "hub" piece, loosen the wheel nuts with a 21 mm socket and breaker bar, jack the car up (you can use the 3/4" plywood hub as a base), remove the lug nuts and wheel, install the spare, lower the car, and re-tighten the lug nuts. Then put the flat tire in the trunk and slide the empty rack over the spare and close the trunk. You can secure the rack with the turn buckles if needed.