Spend $20 on a tire patch kit. I don't carry an extra alternator, tie rods, belts, or any of endless amount of parts they may fail, leaving you stranded.
Knowing your needs, I would just carry a patch/plug kit. Buy a junk tire and practice. Train for the circumstance so you know what you are doing if/when the need arises. IMO, the larger tires look ridiculous - proportion wise - on the rear gate. Also, if you aren't strong enough you'll cause more harm wrestling it on/off the carrier.