I'll start with the usual caveat: CYLOR. The more so since someone local will be more familiar with your specific situation.
It's pretty basic to the definition of a sukkah that it be under the sky. So a sukkah under a balcony, as much of a good-faith effort as it may be (and as much as G-d might appreciate the thought), simply isn't a sukkah - no more so than an Etch-A-Sketch, even one given with the best of intentions, will perform the functions of an iPad.
There might indeed be halachic considerations of which I'm unaware, though, so that alone is a good reason to consult with a reliable authority.
One other factor that might come into play: granted that in your entire neighborhood there are no balconies that can be used for sukkahs (and actually, what about the ones on the top floors of the buildings?), are there any other conceivable possibilities within walking distance? For example, your building might have a courtyard, or perhaps even you could get permission to build one on the roof, or on the sidewalk right near the building, or you could borrow a neighbor's pickup truck and build your sukkah on its bed, etc. (Bear in mind that the footprint of a minimum-sized kosher sukkah isn't all that much; it's about 28"x28" even according to the most stringent opinions, or as little as 21"x21" according to more lenient ones - basically, the amount of space in which one person can sit with a dinner tray. A refrigerator carton would do the job, as long as it's braced so it won't fall over; indeed, one time my father-in-law did just that.) For that matter, your synagogue (again, if it's within walking distance) probably has a sukkah. These are all things to consider before giving up on such a beautiful mitzvah or doing it in a less-optimal way.