It's not entirely a bad bet. From a physics perspective, roughly equal weights of ethanol and CO2 are produced. CO2 is roughly a thousand times less dense than water (this is our reference density=1 gram/mL). (Normal air is 1.2 grams per liter, so I'm rounding CO2 to 1.5 g/L.) At 5% ABV, you know how much ethanol you'll produce. Multiply by the density of ethanol to get weight. And since weight of ethanol almost exactly equals weight of CO2 produced, divide by the approximate density of CO2 to get its volume.
I plugged all that into google as follows:
5% * 5 gallons * 0.79 grams/ml / (1.5 g/L) to gallons
You will produce 132 gallons of CO2. That is clearly enough absolute volume, but the question is whether it will mix too much. There were about three months when I was in university when I would have been able to do that math problem. For all the time before and since, it's been way too hard.
I wouldn't risk it. Is there any reason you can't get a 5-gallon fermenter? They cost practically nothing if you don't mind plastic and are willing to wait until you find something suitable.
* An error in my wording and a mistake in the CO2 density was corrected. Previously I had come up with 195 gallons, which just goes to show that "back of the envelope" math is fairly tolerant of small mistakes!