tmac - thanks!! That's pretty much exactly what I had in mind. As I said, I'm not handy, but DH is and I'm sure he could tackle that project easily.
As for the neighbors, well, the balcony faces an alley where people park, store their trash cans, etc. So I doubt the visuals would be a problem.
And as for the grill, I admit I haven't checked the law, but the house was built about 15 years ago and has a gas hookup built in to the balcony. It seems unlikely that the house could have been bought and sold twice if that hookup was illegal, but I guess it's possible.
Thanks everyone!