It doesn't need to cull all the deer. Wolves would remove the disease for free with surgical precision. Targeting the ill and weak, is what they do all the time.
Or we could vaccinate the deer and let the natural selection act over the Qmamon crew, of course. Even the big birds are vaccinated (and Colbert is a genius).
There is no evidence yet the deer actually get sick. It's quite normal for a virus to be terribly pathogenic in one mammalian species, and able to infect a wide variety of others but without illness or only minor illness. For example, camelpox (related to smallpox) is horrible in camels and kills a quarter of camels infected. In humans and most other mammals it usually is limited to skin sores at the site of contact and goes away quickly. Similarly, monkeypox is highly contagious and usually fatal in its host monkeys, much less contagious and only infrequently fatal in humans, and downright hard to give to a cow or mouse and it won't make them ill. But they can catch it. All of the poxes exhibit that trait, with family- or genus-specific pathogenicity but broad cross-species contagiousness.
This is the reason to humans being unable to do this task, but wolves still are able to discover and target preys with the virus. Preys are proven in the hardest conditions possible. Even a small respiratory problem would eventually appear and be fatal if the prey must endure a long distance chase running from wolves.
Before we had people vaccines, we social isolated. It will be even more difficult to get deer to follow isolation and masking protocols and to show up to a vaccination clinic.