Apfelschorle is a German (or Austrian?) drink which is made with approximately a 1:1 ratio of apple juice and sparkling (carbonated) mineral water.   My technique is simply to pour the appropriate amount of apple juice and water (both chilled) into a cup, give it a stir, and then drink.   When I had bottled Apfelschorle in Germany (primarly two brands: Gerolsteiner and Lift), the bubbles were delicate and small, like in a bottle of champagne. I preferred the Gerolsteiner brand, which had a bit more delicate bubbles and better apple flavor.    I've tried making it at home in the US with an approximately 1:1 ratio of seltzer water (La Croix brand) and apple juice (Mott's brand). The flavor is close to Lift (I suspect higher quality apple juice would give a better depth of flavor and the mineral content is different as well which might also affect the flavor) which is acceptable, but the mouthfeel is all wrong -- the bubbles are typically big and don't last as long as either Lift or Gerolsteiner; only a few sips have the right mouthfeel.   Is there a way to get the right mouthfeel without resorting to buying (expensive in the US) European mineral waters?   I drink a decent amount of seltzer water, so even if it requires equipment to carbonate water at home, I might be willing to invest in it.  
You are comparing two rather different things. A mixture of apple juice and water that is then carbonated (as your bottled product would be) is quite different from a mixture of still apple juice and carbonated water.  I know that both alcoholic and non-alcoholic carbonated cider/apple juice typically have small bubbles. And you can purchase carbonated apple juice in the US, (often seen/marketed as a non-alcoholic alternative to champagne) as well as an increasingly wide variety of alcoholic ciders.  I don't know how mixing carbonated juice and carbonated water would work out, but it would have much better odds of being somewhat like what you want. For even better odds, start with an empty, chilled bottle (suitable for carbonated drinks) and pour both ingredients into it, then seal the top and let them meld in the fridge under pressure for a while (perhaps a day), rather than stirring in an open cup. That MIGHT even work for the still juice, although it would be limited in carbonation.  If you got a carbonation system of some sort, you could start with apple juice and water mixed, and then force carbonate that. 