Be careful with how much "sag" you add to your military tires. Tire manufacturers use several different compounds of rubber in the tire make up, also they add ALOT of extra plies to the sidewall and tread so the vehicle can operate on unimproved surfaces without damage.
Those extra plies make the tire very rigid so even at lower pressures they tend not to bulge.
More modern vehicles have a run flat ring inside to allow the vehicle to get to a repair area so even "flat" there is little bulge.
The ideas and techniques mentioned above are both sound and work well and really do add depth and realism to a model. Have fun, good luck, and show us your finished project please.