Your question seems to be focused on the military impact of fighting/containing the infected population. There are a lot of additional factors that you need to consider to answer the bigger question of can the planet be saved at all? You need to define the behavior of your virus, society, and key decision makers. You also need to factor in interplanetary relationships and the importance of this world in the overall power structure. I mean sure, the US manages to contain the outbreak around Phoenix, but then a new ones are discovered in Denver, New Orleans, Boston, and Seattle. What then?
What is the political structure of your society? Who are the individuals in key decision making positions? What are the bureaucratic power blocks and rivalries? How well do the key decision makers get along? Are the people in key positions willing/able to make the hard call that will piss everyone off? Will the population at large follow the edicts that come out of the government? Or do they scoff at it? How long does it take for the authorities to recognize what is going on and come up with an effective response? Quarantines only work if you know to put one in place, and political leaders (particularly in democratic societies) are often resistant to disrupting the lives of the general population by imposing harsh measures. There will often be an air of denial among the political leaders and the public at large in the early stages of a crisis situation i.e. "things aren't that bad", "you're exaggerating", "that is a matter for the local authorities", "why should I risk my assets/reputation because you were to incompetent to take care of this on your own". The sooner the powers that be recognize the magnitude of the problem and decide they are going to act without worrying about public opinion the better chances they have of containing the outbreak.
You also noted that your infection crosses the species barrier. What impact does it have on the birds? Does it kill them off and we get infected by their corpses? Does it make them more aggressive so that they have more direct human contact? No noticeable impact so there are no clues that they are carriers? Key point here is can they continuously reinfect the human population?
Can the virus infect life forms other than birds/humans? Do people have to kill off their pets? Can livestock become infected? Can people be infected by eating contaminated food?
Are there native plants/animals that might react to it? Would they give any clues as to the nature of the infection? Make it spread faster? Slower? Be a potential cure?
Is there a means of effectively keeping the infected animals away? As an example consider what would have to be done to clear all of the birds out of Chicago, and then keep them out? How would you keep birds from other regions from coming in and breaking your quarantine?
What is the incubation time on the infection? How long can someone be a carrier without obvious symptoms? Is there a way to detect the infection before obvious symptoms set in? Is the medical science of the society up to developing such a test and mass producing it? Is there a treatment/cure for the infection? Are there people who are naturally resistant? Can people be carriers and never show symptoms? How robust is the virus when outside a host? Can it lay dormant and reinfect people at a later time? Are there simple precautionary measures that a layman can take to avoid infection?
The nature of the virus and the political response are the key elements, not the raw number of people.
If we take the rage virus from 28 Days Later as our example. The time from exposure to rage is about 30 seconds. So one infected person getting into a crowded shopping mall becomes 2 after 30 seconds, 4 after 1 minute, 16 after 2 minutes, is pushing 1000 by the 5 minute mark, and over 1 million by 10 minutes. Of course somewhere in there the uninfected are going to notice what is going on and start to flee, which slows the rate of infection. Point being that non lethal weapons have no hope of containing such an outbreak, so it is not a question of whether the security forces can keep from being overrun so much as when do the political leaders admit that they cannot. (and is anyone technically capable of calling for help left among the uninfected when they do.)
If our virus is more subtle, slower, and/or less violent, then other scenarios become possible. But you still have to be able to effectively combat further infections, not the infected population in order keep things under control. If the virus is left unchecked, your society is doomed no matter what the security forces do.
All that said, I can think of very few things that would cause the military to deploy to "restore order" while they are already engaged in an interstellar war. The most I would expect them to do is establish a blockade and quarantine the planet so that the infection does not spread elsewhere. About the only plausible circumstances would be if the planet produced something that was somehow critical to the war effort, AND no one figured out that there was a bio-agent at work.