In the unlikely event that this is a real question, my advice is just to wear it in public in situations in which a costume would be appropriate, such as a Halloween party or something. The fact that you're talking about carrying a phaser to show you "mean business" tells me that the thread is a joke.
Well, the phaser part was an attempt at humor but I am serious about what a general public response would be to some nerd walking into a grocery store and such dressed like Picard.
My mental heath is functioning within normal parameters but I am seriously lacking a "social" life and for me Star Trek has filled that void.
You mention that Star Trek has filled the void of lacking a social life. Just a suggestion, but Star Trek could be used as a way to enhance a social life. For example, there are a lot of nice people here on the boards, and elsewhere who share the same interest. Perhaps a meet up, if there are others here in your city. There are fan clubs, conventions, etc. My best friend and I met in 1986 at a sci-fi convention. Just a thought.