Do you think it's bad if I tell someone I've known for less that a month I don't want to be their friend because of their mental health issues?
What about their mental health issues bothers you?
They seem very unstable and display erratic behavior. I'm afraid of what could happen when I'm around them.
This sounds like a difficult choice to make, but I can see why you don't want to be around them. Try to find a way to tell them gently so that the news doesn't trigger a negative reaction from them.
I don't have to worry about their feelings, I can do what I want.
This person may have a very fragile mental state so it's important to treat them gently. Stopping your friendship could be traumatic for them and really upset them.