Oh yeah, you can totally eat pigeons, and people do. You can totally eat rats (and I assume some people do). But those aren’t the ones you randomly spot eating garbage in the cities.
I remember a tour guide in Paris showing me a poster from the Siege of Paris by the Prussians.
Dog meat, pigeons, rats, etc.
Honestly I’d just do what Napoleon III did and just sit in the field and wait to get hit at that point.