I'm not sure if labelling food as genetically modified is a useful warning, or if it would be the same as saying "wifi used in this building". WIFI isn't harmful, and eventually it's going to be everywhere and not avoidable. Yet you still have people who claim that electronic smog causes all sorts of things.
I'm happy to eat these things. I'm gently worried about releasing organisms into the wild with "exogenes" (or whatever they're called). Evolution is amazing and powerful and wonderful. And human intervention in eco-systems isn't filled with particularly great examples - a long list of invasive species comes to mind.
And I think I get your concern about the labeling as well. You're worried that people will assume the worst. I mean, you called it a "warning". But I expect it to be more like an organic labeling. Some people will care, most won't. Is it useful? Only to those who care.
I'm already seeing a number of products proudly claiming they don't use GMOs.
Edit: (sorry for changing my comment, I add the original below).
You are picking on the wording. There must be a succinct way to say non-genetically modified animal.
If it's GMO: GMO-Chicken. Seems succinct as well.
Non-GMO Chicken. Least succinct of the 3.
"Has excrement" would certainly be perceived as negative, but the effect on my shopping of "has no excrement" would be a lot more negative (trying to find this sticker on everything I purchase, for example).
If informed people won't do something, the answer shouldn't be to keep them in the dark, no matter how much you disagree with their reasons.