What can I do about a menacing charge?

Unfortunately, the beginning of the criminal process can be as minimal as you describe.  If someone files a complaint and the cops believe, they can file charges without ever interviewing you or asking you to give your side of the story.  The minimal requirements needed to file a charge and potentially ruin someone's life are probably one of the most glaring flaws of our justice system.
On that note, however, the minimal filing requirements do not necessarily mean that is all that will be required to convict you.  The standard for review is much higher. 
This issue you face right now, though, is how to deal with the process in between.  You have a couple of different options.  The first you have tried to invoke.  So many people invoke their right to remain silent, that law enforcement often assumes that you don't want to give a statement.  If you are able to give a statement and present some evidence to the investigating officer, they can forward a supplemental report to the prosecuting attorney.  It looks like you've already tried to invoke this.... but it's still an option but you may be able to invoke.  You can hire an attorney to help arrange and interview and go with you to give a statement.  Often, local defense attorneys will have a direct line to many of the investigating officers and can be a bit more successful in facilitating that meeting.
Your second option is make an open records request directing to the investigating agency.  This won't make the case go away... but at least it will give you an idea of what they are basing the charges on.  If you have a friend that is willing to make the request for you, you will have a better chance of getting the report while the charges are still pending--- because sometimes they don't like giving reports to defendants prior to court.
You can also go to the clerk of the court and request a copy of the documents that are currently on file.  Sometimes the court's file will also include a copy of affidavits used to support the charges against you.
So... the previous steps were more about gathering info... after you have a better idea of what you are facing, you can start working on shutting down the charges.  You can attempt to represent yourself, but you run the risk of improper evidence being used against you or you making a statement to the prosecutor that could be used against you later.  If you cannot afford an attorney, talk to the court about obtaining a court appointed attorney.
You mention filing a complaint for defamation of character and false report.  These are options that you can invoke.  However, visit with a criminal defense attorney before you do.  Normally you cannot be compelled to give evidence against yourself... .however, if you file a defamation suit, then you could potentially waive that priviledge.  This means that the prosecutor would have access to the same info if the defamation case was pending at the same time your criminal charge was pending.  It's usually best to wait until after the criminal charges are resolved to file a defamation suit-- unless you start to his a statute of limitations issue.