First of all, my sympathies to your mom, that sound very painful. Since I was an omnivore for almost 50 years I don't have that much of a problem cleaning up after meat eaters although it disgusts me, but I can understand if it really bothers you.
However, your post hints that there is more to this. That you may be feeling like you are bearing an unfair portion of the responsibility to help your mother and that possibly this was the case even before she was injured. So you appear to be making a stand on the dirty dishes issue, which unfortunately transforms it into being about your veg*nism. If that is the case, perhaps you could address the issue from a different angle. Explain that it's unfair for you to have to do all the work while your brother doesn't do anything. If she is receptive, then you might suggest that he could start by washing dishes. It's worth a try, although it sounds like your mother is reluctant to make your brother do anything. Good luck.
By the way, I do get this kind of thing from time to time. I was making sandwiches recently and asked my daughter to put the meat on her sandwich if she wanted it because I didn't want to handle the meat. This was interpreted as me "forcing my veg*nism on her and trying to make her feel like a bad person". It wasn't, I just didn't want to touch the meat, I thought she'd understand, but she doesn't always see things from my perspective.