Please don't. Having been on both sides of the table, and having seen this song done with and without a prop, I can honestly say you do not need it.
When I have seen the song done with a notebook, the auditionee's focus has really gone down to the notebook, we really lose the person, and the auditor needs to see you! Also, I have seen some awkward handling of the notebook, because you don't have a desk to put it down on, so you're trying to balance a notebook, "write" in it, and sing your song. It can be distracting.
You can always approach the song as laying out the letter prior to writing it, verbalizing it so it sounds "just right" on paper.
Off the top of my head, I can't remember a favorable audition with a prop, and would not generally advise using one.
Just my two cents.
A man is getting along on the road to wisdom when he begins to realize that his opinion is just an opinion.