Earlier quoted context omitted.
There's no problem mailing dead animals: > The following factors are applied to all shipments of mailable live or dead animals: > … Protection of the mail and the environment against the following: … Obnoxious odors and noise. > ...protection of animal specimens against spoilage https://pe.usps.com/text/pub52/pub52c5_003.htm There's some other restrictions on mailing dead wild animals, but mailing livestock carcasses…
I can't tell if you're being intentionally dense. Lets do a different example. Lets say I go to checkout at a grocery store and I ring up a bar of soap as a bag of ice. Obviously this is wrong. Nobody is debating you can buy soap at the store but to claim it is ice is wrong. Back to USPS. Nobody is claiming you can't send live animals. The question is what happens when you send a live animal and don't claim it.
There's no need to declare anything in that case. Might need to declare the preservation chemicals, but nothing about the actual dead animal.