Having gotten 3rd degree burns myself over 20 years ago, I have enormous sympathy for you and all other burn victims. I had no idea how bad pain could be before then. I'm not sure I even remember now. It's unspeakably horrible.
A trike might be the ticket for you. How much use do you have of your burned side? Trike brakes can be set quite sensitive so you may or may not need self balancing brakes. I typically use only one or two fingers to brake on my trike and you never need to squeeze hard like you do on other types of bikes.
My recollection is that the pain is significantly reduced if you keep the injured part elevated and use it relatively little -- the riding position and characteristics of a trike are good in this respect. A cat is a good choice because they are light which will make them easier for you to move around. Whatever you get, make sure it has bar end shifters. They'll be easier to operate because you can operate them by bumping them or pulling quite lightly.
Your burns are worse than mine were, and while some of the damage is undoubtedly permanent, things will improve but it takes a long time. Hang in there.