If you cooked the dead rat, and the dead chicken, I'm pretty sure they would not be toxic. Unless the rat had just eaten a lot of rat poison. Raw might be OK, too, but I'd worry about disease.
People have survived on worse foods.
Once those chicks start to die, the postmaster will almost certainly lose their job!
When authorities step in, the USPS will not want a "black eye"- they have enough problems! The postmaster WILL be fired!