So you mimicked your brother's sickness to a someone else and now feel guilty about it. When you went home and told your mum, she did not seem sympathetic, and I guess you feel worse?
There are a few ways you could process this...
(1) do nothing and work through it
(2) apologise to your brother
But whether you do (1) or (2) depends on your motivations for the way you spoke (or not). Initially you said that you wanted to make the conversation lighter. Now, rather than carrying on if you had stopped before going too far would you still feel guilty?
One thing I can say is that God will not make him sicker. I would think that God is with him at the moment, in his sickness.
Perhaps your attempts to lighten the situation, or to remove some of the distress you feel about the brother was to copy him? Does not make it right or wrong, but at that moment in time, that was how to you dealt the situation - I can tell from your post that while you feel guilty (which is also a natural human feeling), there is also a great concern for your brother, which is much greater than what you said. This is also shown by you feeling the need to post here.
Now I don't know how old you or your brother is. but an apology could strengthen the relationship (over time). You are not the first or the last person so do something like this... in fact, I probably did this to my brother and he to me many many years ago. It does not make you a bad person.
Might not have answered your questions or gave you any advice, but if you want to chat some more about this, I am here and will listen to you.