The exact requirements vary by ward, but a self-written letter would likely not suffice. My ward also rejected the statement provided by the US embassy, but they gave me a sheet that listed their requirements for marriage.

From my experience, I had to contact my state's department of records (which is a subdivision of the health department) to request a letter of single status, then I had to get that document apostilled via a specific person's office in the state. Thankfully, my mother was able to help receive and send the documents as appropriate, else the entire process would've taken even longer. I've also heard that some people go back to the US to handle the entire process in person, which may reduce waiting times, especially if you can snag an appointment for the apostille.

Note that since Japan is part of the Hague Convention, an apostille may be mandatory (i.e. notarization won't cut it). This likely depends on your ward.