Actually there are "pet" CITES permits issued on the US side that allow transport of pets back and forth with one permit. I have assisted customers and friends with obtaining these in the past. The one stipulation they will ask for is that the animal in question be microchipped with a chip that is readable by both border crossings. As well the animal may and only be transported through one border crossing. This was as of last year and since then the system may have changed. The best way is to just call them and ask or email them from their respective website.
If everyone could just get along this world would be better for all living things.