The chemistry you're describing the the one you expect: CO2 is formed, and released from the liquid. However, your question is more about kinetics than about the final product. We can assume that slow release of gas leads to larger bubbles, because nucleation rate for the bubble formation is lower. Now, in fizzy drink, there is a large quantity of carbon dioxide accommodated by the liquid due to pressurization. In your case, it will depend on the concentration of your lemon juice (natural lemon juice has pH ~ 2), and more importantly on that of the carbonated water: I just tried with a few grams of sodium bicarbonate dissolved in a glass of water in my kitchen, and the kinetics and bubble size really depends on initial carbonate concentration. (I haven't been able to make pictures or video, because I had to handle my kids who were very fast drawn to this impromptu experiment.)
Also, I second Richard's comment: there are many additional chemicals in lemon juice, some of which surely act as surfactants… which changes bubble formation and kinetics.