I can't see any reason why running smaller tires would be an issue. I think the weight rating is key here. Theoretically, your fuel economy will increase because the tire/road contact surface is smaller, but it's probably too insignificant to notice. The same is probably true for the decreased braking effectiveness. But that's just my guess.
I'm not a gambling man, but I
would place money on a bet that you would not check the air in the spare before each trip.
Mine is mounted on the rear bumper, isn't covered, and I don't check it.
Dave