Definitely allow it. Then check back in every three or four sessions and talk about how they're connecting with their roots, and feeling the slow flow of sap through their limbs. Maybe do a little role-play with an elf minder who swings by to fumigate for bugs and sing their character a song or two. I'm also in the camp of "becoming an ent is a decades long process".
For that matter, I never thought that elf death was any different from anyone else's death rules wise. They're dead and out of the game. The fact that their spirit lives on and can form a new body is irrelevant since I believe that process is years long, and beyond the scope of any campaign I'm running. Still, there is plenty of story elements to make this ability interesting I think. An elf could very easily be newly reformed, having been lost for centuries before being allowed to reform their body. Such a character might have totally missed the Blood Mist, or even the arrival of man. Or perhaps they fought Zygofer's demons the first time around, or were friends with Martea before he turned her.
Immortality has some fun story elements, especially when you could have been killed, "lived" as a gem and gotten carried around for a few decades, been reformed, fought invading humans as an ent, died again, been lost in a river for a hundred years, found by a halfling while he's out fishing, brought to an elven druid to be reformed again, and ultimately start your new adventures with that halfling's offspring.
Anyway, definitely yes to an elf becoming an ent. But definitely no to anything that breaks the game like characters being monsters.