We have done hoop houses and covered a baker's rack with 6 mil here in the DFW area. I had to monitor weather and open/close the ends and even call my neighbor to ask that he open the ends on warm days when I was at at work. I wasn't overwintering epis, but I did have tomatoes in there. It was general house tropicals and veggies (that looked great but did not set fruit).

I'd use the thickest mil plastic you can find and use gorilla tape to hold it up. I'd also go ahead and let the dryer vent into the patio for warmth and moisture. Remember that you really only need to keep the temps in there around 50 to protect those plants so it shouldn't be too hard in your area.

Another option you might consider is using a metal utility shelf unit in your kitchen or bath and cover it with thin plastic sheet. Set a humidifier on the floor under the unit and provide grow lights if its not near a window. This is what I used for a couple years to start my veggies indoors before I finally got a greenhouse.

I get heavy vinyl, sold in a big roll, from Hobby Lobby to staple to my greenhouse. It works great - but to fortify it, I got a roll of black mesh about 3" wide - a local nursery got it for me - to put over the plastic - this keeps the staples from ripping the plastic. You could staple this to roof of your balcony, leave enough to put bricks on it on the floor and raise it on warm days. I love this vinyl - also use it to cover the windows on my chicken coops. You can buy several different sizes of greenhouses that would fit on your balcony - Big Lots has them, Atwood's, Tractor Supply.......maybe some other stores, possibly Lowes/Home Depot. Have fun!

when Jim lived by Baybrook mall we bought a big roll of plastic from Home Depot and enclosed his patio. All the plants did fine, no supplemental heating and we didn`t do anything on hot days. The dryer vent might be a bit drying for the plants so just check the moisture level of them. The main thing for us is just protecting them from the wind.
Tally Ho!