hyde, sometimes rubber tires work better than foams on concrete parking lots. it really depends and only answer is to try. loose diff might help as it acts like a slipper in bumpy conditions to maintain traction.
JayL, maybe the hook where crane picks up car is not centered and more to the rear, hence nose down when picked up. i would guess weight is 40/60 front/rear. engine/transmission/bigger tires/driver are all towards rear.