Player RoninX over on the Steam forums has diagnosed and figured out a workaround for this problem.
The problem:
1. If you have an animal whose Species Name is longer than the available letters, such that the species name is getting "..." added to the end of the species name in the trade center....
2. And if you've got a species with one of those long names up for sale (such that it's showing "xx mins"),
3. THEN, you cannot put up for trade any other animal, regardless of the length of their species name, as long as at least one animal with the "..." and "xx min" in the same line is still showing visibly on your screen.
The solution is indeed to scroll down, or re-sort your animals in such a way that no animal that has BOTH a long name and is already up for sale is showing visibly in you window.
Props to RoninX for finding this. It took quite a bit of detective work to diagnose, since everyone's different language settings meant that different species were causing the problem for different players, and because one of these animals being visible in the window makes every other animal above and below it untradeable until you've scrolled the offending animal off of the visible screen. (which meant that people were reporting the problem quite differently depending o what happened to be in our trade centers, and what happened to be further up or down the window from the animal we were now paying attention to).