Slight Problems: 
1) the name sounds kind of like you are a cheap knockoff or a sequel.  You could add some color or call yourself Black Bat or something to distinguish yourself.  For example, my GL character is nicknamed Little GL to distinguish me from the other GLs.
2) I'm not sure if Tim Drake exists in this universe.
3) you would have been four when earth was sterilized, so you were depressed over it most of your life?