If you are dosing gas CO2 than you only should be doing it during the photosynthesis period. If you are dosing liquid carbon products, then they can be dosed day or night.
In planted tanks, bubblers exchange too much oxygen for CO2. Which of course you have identified as a more oxygenated environment for your fish. We don't run bubblers because they are unnecessary. All that is really needed is the water surface being rippled which promotes oxygenation in itself without the heavy exchange from tank bottom to tank surface. You do need to have enough surface agitation though. This can be done by using your power filter is you have one and allow it to cascade into the tank instead of the out-spouts already being below the water surface.
Many things govern whether CO2 injection is necessary or not. I have a moderate to heavy planted planted tank with no injection. Why? Because light wattage per gallon is the factor. I'm running about 1 watt per gallon. This means my plants are low light and slower growing plants. I don't need CO2 injected. I can get away with dosed carbon. Now that's not to say CO2 wouldn't be beneficial. However, if you're running something other than a low wattage tank, then CO2 injected becomes more of a requirement.
Here's a video I just took of my tank to show you my fish aren't needing a bubbler. They all swim from bottom to top and really only surface to eat and play in the stream of forced water on the right side. My corydoras never breach the surface and always stay submerged which tells me there's enough oxygen.
Pardon the background static and poor video quality. The mic is way too sensitive because I have no idea what ambient noise it was picking up and amplifying like that.
http

/www.youtube.com/watch?v=JiYdr9tWcog
Also what's sparsely planted? Can you give us some info on your tank, light wattage, and how many plants you have?