Spare Tire issue with bigger tires?

Yesterday I installed Cooper Discoverer 245/60 on stock 18" wheels, but a question comes out: if I ever need to use the smaller diameter spare, it will damage the SHAWD system? This because a couple of days ago, got a flat on the right front, used the spare, it had low pressure, so I drove a few yards by making a U turn, ( steering felt weird BTW ) and warnings for ABS, VSA, SHAWD showed immediatly, I entered a gas station, set the spare tire pressure to 60 psi, drove out, lights went off.
Another related issue: the only warning didn't show was the TPMS after the flat was in the trunk...