Everyone's situation is unique, however, they aren't likely to turn you down unless you show signs of self-harm or other serious and unaddressed mental-health issues. Like it was previously mentioned in this thread for many of us our stress/anxiety/depression has firm roots in our physical health and self-image.
Full disclosure: I'm in a similar position, currently on Cymbalta. After my Psychological testing and interview I was permitted to have the surgery on the condition that I continue having therapy.