I think he wrote them because his sister had the same amount and kind of information as what Anne had in the note, which leads me to believe he must have written the note.

Also, if the note were from Mary (whom we already know is not really one for bothering to write to her family and was having too good a time in Lyme to have bothered, probably)--why would Anne not recognize that quite easily, or think it a mystery how it got to Kellynch lodge?

Also, would Anne have any reason to recognize Frederick's handwriting? When they were together, they were together. He wouldn't/couldn't have written to her until they were engaged. He didn't go away anywhere during the short engagement, and he sure didn't write to her after it ended.