Nothing wrong with shooting a doe fawn.
It's a doe, and if your management program calls for removing does, a doe fawn fills the bill nicely. And it will save you money....you won't have to feed it at your protein feeders or food plots to raise the fawn up to adulthood, just so you can shoot it next year. It may seem cruel, but it's somewhat better for your management program to choose a doe fawn over its mother when culling.
Just be certain it's not a buck fawn.
And I'm putting my money on pneumonia as well. It's quite common in deer, sheep, and other ungulates. Rabies would be highly unlikely, but it's not worth taking the chance.