It is an interesting theory to consider. The writers did say that Belle would play a bigger role in future episodes, and Belle was said to be a lover of books, both in the Disney film and the actual fairytale. It would be another good node to the original stories, and would give Belle’s character a larger purpose.
Also, we don’t really know how long Belle stayed at the castle–I’m inclined to think it was a decent amount of weeks, maybe more. Maybe while she lived with Rumpelstiltskin, he told her a few stories of his dealings and she wrote them down. It wouldn’t be too far-fetched if they both talked about some of the people he met and what he did, and it wouldn’t be hard to imagine Belle writing about it.
Yet she did leave the castle after many of the events happened, so not sure how she could have still wrote in the book. The option of the author being Rumpelstiltskin is much more stronger–his story does not seem to be present in the book. Neither about his son, nor about Belle–probably because he doesn’t want people to know about them.