I'm going to cut out my mentally ill friend
That feels very sudden. Why would you want to cut them off?
She's too much of a bummer to hang out with
I understand that hanging out with someone who is mentally ill could be a bummer sometimes, but that comes with the friendship. You should be there for the person for support instead of always expecting them to be happy and cheerful. If you don't want to be a source of support for the person you shouldn't have been friends to begin with.
Being a support sucks.
I agree that you need to exert more effort being supportive with your friend specially if they have a mental illness, but it is not right for you to think of it as if it is a burden you have to bear, as your friend may not have a lot of people that can help her during the times they need you the most.