I think it would bubble off too fast, so you would have a good level of CO2 for a few minutes, then the plants would use it all, or it would out gas. You would have to dose it every few minutes all day long.
Anything you get to auto-dose usually will have the carbonated water exposed to the air, so it will lose the CO2 anyway.
Even if it was easy to pump in the carbonated water and create a constant supply of it, it doesn't sound very healthy for the tanks inhabitants. Bottled carbonated water is not just water with carbon dioxide in it; it can also contain salt (NaCl) and brands that list TDS are near 1000 ppm. Having said that I have found a supermarket own brand that looks ok so I think I'm going to have a go with that when I get back from America.