There's some other restrictions on mailing dead wild animals, but mailing livestock carcasses is not restricted from what I can see outside of those broad guidelines.
I believe the Hot Zone (though maybe not the Reston incident it was based on?) starts off with leaking deceased monkey remains arriving at a pathology lab.
Few better ways to spoil a pathologist's morning than a package with a blood stain in one corner.
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.
> 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 is not restricted from what I can see outside of those broad guidelines.