Maybe you should try a different chair. I don't think they can complain if you provide your own, or provide your own back support for the awful one you have. some find that a kneeling chair is an option, as it changes the way you sit entirely. They have ones with and without back supports.
the thing is going to be to find a chair type that is comfortable for extended use and to get a doctors note saying that you need to use it for medical reasons - then your work has to allow it, vs the one they provided. Then you can bring yours, put your name on it, and no one else can use it.
I have a kneeling chair which I don't always use because it does make my legs sore quickly but I feel like I play better in it than a normal chair
I'll wear a lightweight scarf, loosely hanging down the front, which kind of obscures the view. Seriously, whenever I've noticed someone staring it's always a woman with a judgey look on her face!
See if you can find a chair that looks like this to try. If it seems comfortable and your boss won't buy it, ask if you can get one to bring in.
I can see that being comfy (or comfy enough to forget).
It reminds me of my Dad's computer stool a little.
I have a kneeling chair that I love. Mine is from the 80's, but it's something like this and it is surprisingly comfortable.
If you're a fan of good posture, one of these works great, but if you get one invest in some additional padding or you'll eventually hate having sore shins all day.
I got this one: https://www.amazon.com/gp/product/B000TMK0O0/
And later got this cushion for it, bc the padding just isn't enough: https://www.amazon.com/gp/product/B01EBDV9BU/