There really is not a problem. Best tool of human beings' advancement is the emotion of inferiority and superiority both. And people are compelled to tell you on their opposition to not only show you, but to themselves that they are better.
The problems people have you hunting for deer meat are not a single reason. Some might have religious oppositions whilst some might be against it purely for "animal rights". If they are against your hunting and consumption of deer meat because of religion (which i highly doubt), they would probably not really have a problem with you specifically, but consuming it themselves.
But if it is due to reasons of "animal rights" or "animal loving" their motives are due to the emotions of interior/superiority, unless you have hunted their pet deer and ate it.
Any modern movements supported by trends will have numerous followers who define themselves by their "importance" and "knowledge" of what society should be. Everyone thinks they are right, no one thinks that they are doing the wrong thing. And this is perfectly natural.
However, self-righteousness driven by a trend is a serious problem. This creates an almost fanatical preachers who are almost always extreme, in seeing things very black and white. As long as people do not give you problems, their opposition is respectable as it is their opinion.
Vegans, Vegetarians, Animal lovers, pet owners all have their opinions. But they should not use you, as a tool to feel better about themselves.
p.s. I've never eaten a deer... but now I really want to try.