Not Bathilda Bagshot
There is no mention in canon of who the author is. (You can find a nice compilation of most of what we do know about the book on the HP Lexicon). None of the places where the book is mentioned in the series, interviews, or any other canon source provide the author.
However, we can probably rule out Bathilda Bagshot. Every time that Bagshot appears, from her being mentioned in casual conversation to her review on the back of Quidditch through the Ages, she is always only identified as being the author of A History of Magic.
Particularly telling is Hermione's association when she hears the name in book seven.
“Well then,” said Harry, keen to discuss his theory, “what about information on Dumbledore? The second page of this letter, for instance. You know this Bathilda my mum mentions, you know who she is?”
“Bathilda Bagshot, the author of —”
“A History of Magic,” said Hermione, looking interested. “So your parents knew her? She was an incredible magical historian.”
Harry Potter and the Deathly Hallows - Chapter 10
Remember that Hermione was a big fan of Hogwarts: A History to the point that she carried it around wherever she went even when it didn't make much sense.
“Here he goes,” Ron said to Hermione, rolling his eyes.
“As we knew he would,” she sighed, turning back to the books. “You know, I think I will take Hogwarts, A History. Even if we’re not going back there, I don’t think I’d feel right if I didn’t have it with —”
“Listen!” said Harry again.
Harry Potter and the Deathly Hallows - Chapter 6
If Bathilda Bagshot was the author of Hogwarts: A History then that would have been the book that Hermione associated with her.