I’m very sorry to hear about these developments with your mom. It can be devastating to watch people we love develop debilitating problems. Give yourself the benefit of not fighting with her anymore. She’s convinced, and people who have this strong internal narrative are hard to convince.
There are some studies suggesting that paranoia in the elderly may be something that gets activated by hearing loss and or distortion. It’s interesting that she believes you need hearing aids when the actual issue may be just the reverse. Tell her you want to help her get to the bottom of this and the first step is to understand what she is hearing and to see the sources.
I would begin with a hearing evaluation. This is often less threatening than saying she is paranoid. Under the general umbrella of trying to help, offer to go for a hearing test for yourself and with her. This may help as you are saying that you are willing to follow up on her suggestion by getting a hearing test. You may want to also have a discussion with the audiologist separate from being with your mom about the fact that her hearing seems to be distorted. If you offer to go with her and expose yourself to the same hearing test — this might give her enough support and confidence to see the audiologist.
The audiologist can then make some recommendations. This may be helpful in getting her in front of my psychiatrist or other mental health professional if that is needed. Since the problems have begun as a result of hearing things — and she has accused you of not having adequate hearing — this might be a way for you to help her begin dealing with the paranoia. The most difficult part here is the fact that her thinking is confirming her distortions. Losing your hearing can be very difficult phenomenon to cope with. Here is an article about hearing loss and paranoia in the elderly you might find interesting.