I'm not sure which cruise line you are booked on; it doesn't seem fair that they would charge you an extra $200 for a disabled cabin. How much proof are they expecting you to give them? I understand that the disabled cabins need to go to the passengers with disabilities & you are one of those passengers, I think.
I just booked a cruise on Norwegian for my in-laws 50th anniversary. We have seven cabins booked for all of our family members. I had mentioned to my cruise assistant at NCL that I thought my in-laws would do better in a disabled cabin as my mother-in-law has had both knees replaced & very bad vision due to a tumor. YES she can walk, but not far distances. YES she can see, but not very well.
Based on what I told my cruise assistant, my in-laws were assigned a disabled cabin. There was no up-charge & the cabin is guaranteed. It's unfortunate that there are people out there that will make up any story to get a larger cabin, especially when there are people like yourself that really need that disabled cabin.
Maybe you should try calling the cruise line directly again but ask to talk to a supervisor. Hopefully, they will understand your needs & give you a cabin that will allow you to enjoy your cruise. They need to understand that people CAN be considered disabled, even if they don't use a wheel-chair.