I'm still a newbie/AFC myself, but I'll try to help. First of all, you're right about smiling being important when you approach a girl. Place yourself in the shoes of someone who is being approached. Would you rather be approached by a girl who has a big smile on her face, who looks happy and fun to talk to, or a girl who comes up to you with a straight face as if she's playing a game of poker and is trying to give you as little information as possible?
You could always try to think of something that you think is funny. Just spend a few minutes thinking about funny experiences you've had, funny movies you've seen or funny jokes you've heard. Anything that brings a smile to your face whenever you hear it. And whenever you need to smile, like when you're approaching a girl, just think of that thing/those things that makes you smile. One doesn't need to be in a state of bliss to have a genuine (looking) smile
I also think you need to work on your inner game, being happier with your life and who you are will definitely help you in being able to smile when you want/need to.
I don't know much about inner game myself yet, but I'm sure someone else can help you with that. The only thing I can say in regards to this is: try to find out what is making you feel anxious and depressed, and do something about it. Try to think of all the good things about yourself, e.g. what makes you stand out from most other people. Focus on these things, and try to get out of your state of anxiety and depression.
Also try to 'warm up' before you approach a target. Try to be social with other people, just talk to them and get into a talking/being social mood. I think this will help you with smiling when you're approaching your target as well. Already having talked to other people (assuming you do it in a good way) should make you feel better and more relaxed.
And finally, work your way up. I can't really relate to your problem, and I probably don't really know how your situation really is. But try smiling on command with family and close friends first. Whenever you're able to do that, move on to the rest of your friends etc etc etc. until you're able to do so comfortably with everyone you know. And after that, just try to socialize with people in the club and see if you can pull off a genuine smile when you approach strangers. And when you're able to do that, you're ready to try again with your target(s). Sometimes taking small steps and working your way up is better than trying to immediately do what you have set as an end-goal (in your case, being able to pull off a genuine smile when approaching a girl).
I hope this was helpful to you, good luck with it.