I'll relate a story about a guy I know, except this dog wasn't chasing deer, but was on this guy's property, unleashed, after his dog had been mauled a few times in the past. Dog wouldn't respond to his yelling at it and continued walking toward his house. After waving his arms and yelling at it, the dog continued without responding to attempts to scare it off, so he took a blunt and shot it. Turns out the owner was nearby too (unseen). Dog lived, but only because the property owner bypassed his broadheads and chose to use the blunt, cuz he didn't want to hurt the dog.
Owner was cited for trespass and unleashed dog, but the property owner was hauled in to jail and is awaiting trial... and this was for protecting his pet and family... Not to mention the harassment, tv stories and paper articles, loss of business, death threats, vulgar voicemail messages his grandkids heard...
You make your own decision if it's worth it if the dog is just running deer.