Take with you any documentation you can about your mother's condition. Someone mentioned her right to privacy. I wouldn't worry over that. Your right to privacy has already been sacrificed. Call your state level gun rights organization. See if they are able to offer any assistance. They may have attorneys who work cheap for their members, etc. As others have said, document that you can safely lock up the guns where she has no access. If you have no type of safe or cabinet and can't afford one, consider putting locking hardware on a closet door and use a secure combination padlock. With no keys to potentially float around, she can't gain access. If you do have a cabinet, put an extra combination lock on it.
Cops are between a rock and hard place with situations like this. Everyone seems believable but obviously someone is not truthful. They don't know any of you. I use to do some work and studies in a mental hospital. Mental patients can seem perfectly stable and sometimes are for a while. They can convince casual observers nothing is wrong with them.