To be sure were on the same page.
Are you thinking of just carrying around a tire minus rim and then change out the tire onto the rim?
If thats what your thinking like Will said it would be way easier to carry around a spare tire mounted to a spare rim.
Changing a spare tire on a HMMWV thats already mounted to a split rim is no different then changing a spare tire mounted to a non split rim. You will only require a lug wrench and jack.
As to changing the tire out on a split rim it would depend on if your running a run flat in the tire if you are then yes you will need some more stuff. To do that on the side of the road would be a bitch without some special tools.
If your wanting to unbolt the 8,12,24 bolts that hold the two half's of the rim together and remove the tire off of it and remove the run flat out of it and install the run flat back into another tire then it would be wise to carry around a spare o ring and run flat compression tool, soap, and a vocabulary that would make a pirate blush.
Not sure that helps .
BH