Quite often supervised visitation only lasts for a few months. Seldom does it last for over a year unless there are real issues of drug or child abuse. I don't know how long yours has gone on, but if it's been awhile, most judges would be open to changing it absent any reports of problems while such visitations were occurring.
Have your attorney file the necessary papers with the court to have the visitation changed to unsupervised and ask for a visitation schedule that is customary in your county.
I do not believe that a judge will believe that someone who is classified as disabled is somehow unable to handle child care responsibilities on an occasional basis, such as every other weekend, especially when another adult is likely to be present. Many people who are collecting Social Security Disability Income, for example, do have children. They manage to take care of their kids, usually on a full-time basis. Many of them are single parents as well. Of course, the nature of the disability could have a bearing on this issue. But, going into court, a judge is not going to automatically think that someone who is disabled is somehow a danger to the children or otherwise not capable of normal child care responsibilities.