There's a way to root the Bionic on JB. However, it can only be done on Linux (the instructions given are for Ubuntu, which I imagine would also work on Mint). If you don't have Ubuntu installed, you can run a LiveCD (basically running the OS without installing it on your hard drive) and perform the process. I've played around with Linux and the terminal, so I wouldn't have a problem doing this but if you never have before, it can be overwhelming.
I'm undecided if I'll try it or just wait for a .23 update