You're reading too far into it. The Ents were spirits of good and wouldn't have fought for Sauron, similarly Orcs have spirits of evil and wouldn't have fought for the Last Alliance.
Let's nitpick the quote.
All living things were divided that day, and some of every kind
Tolkien seems to be using what's commonly called "A figure of speech". Every living thing may have been divided, but whether or not every living thing was at the battle is questionable. They may have been divided but not present.
even of beasts and birds
Ents seem to exempt from this being neither beast nor birds.
were found in either host, save the Elves only.
This seems to be suggesting they were fighting on either side, but again I think Tolkien is using an exaggeration to increase the "impressiveness" of the battle.