Of course all emissions should be counted. It's not just the explosions and burning oil, I'd guess that manufacturing all the steel and chemicals also uses loads of energy. Some stockpiles used now may be associated with emissions long ago, e.g. in the last decades of the soviet union emissions rose very high, even while the economy was low.
That got me interested on fuel economy.
According to this webpage, a M1A2 has a gas tank size of 1907 l (505 gal) and a cruising range of 426 km (265 miles).
That would make 448 l/100km (0.52 MPG). Wow.
The site also says
A tank will need approximately 300 gallons every eight hours; this will vary depending on mission, terrain, and weather. (1364 l)
0.6 miles per gallon.
60 gallons per hour when traveling cross-country (263 l)
30+ gallons per hour while operating at a tactical ideal (136+ l)
10 gallons basic idle (45 l)
A mine plow will increase the fuel consummation rate of a tank by 25 percent
So that suggests, over 4 tons CO2 per tank-refill. Many of those things don't get to roll very far (except by train, ship), but there's still over 120 tons embodied CO2 just from producing the (mainly) steel. Also the energy in the shells.
I guess military planes, ships, missiles contribute more than tanks. Should also consider albedo effects such as smoke drifting over arctic snow.
But maybe this is all dwarfed by the implied emissions of reconstruction later, also missed opportunities for cooperation on global mitigation efforts.
Probably you are right with the latter. A cement brick house easily has 100 tons CO². And in war, whole cities get destroyed. Plus destruction of enemy energy infrastructure, like oil fields, if existant.
Kind of sad now, when I think about it. Looks like we rather destroy the enemy with us, than having somebody we don't like rise above us.
causing houses to be abandoned, necessitating houses elsewhere while the abandoned ones likely get bombed
decreasing the number of future consumers, whose future footprint would depend on future behaviour patterns (hard to predict)
changing future land use patterns, either due to unexploded ordnance or straight out chemical contamination (there are places in France that are still off limits to economic activity, because World War I contaminated the soil with toxic chemicals), here in Estonia there are still forests from which you don't want trees in your sawmill because they contain shrapnel and bullets from World War II
I have the feeling that calculating the climate impact of actual war is a difficult job.
But they could calculate the tonnage of spent fuel and energy, that would be easier.