Seems like a masochistic way to go about adding CO2.
You could do it. Adding carbonated liquid to a non carbonated liquid avoids the 'foam up' that you get that you'd have if you added a non carbonated liquid to a carbonated liquid, keeping more of the CO2 in solution. (Add champagne to orange juice, observe fizz, add orange juice to champagne, notice that you get less fizz).
If you made a large bubble counter on an existing yeast CO2 system, you might be able to make your own carbonated water.
Its just a novelty however. Not practical. We could buy our own hydrogen and oxygen tanks and make our own tank water, but there's better ways to go about it..
You could use a container of baking soda, and set up a drip system of vinegar to make CO2 as well..