If you’ve got a carbonation setup at home that you use for your homebrew, cider, mead, or kombucha, you certainly can fizzy up some wine products for yourself. I like your idea of using a dessert wine, because the sugar in sweet wines can balance out the sensory “sharpness” of bubbles, possibly leading to a very harmonious outcome.
I think you should experiment a bit with the level of carbonation; starting low and turning up the pressure gradually can be a good way to come to the perfect balance point. The first thing to do is to think about your serving temperature. The laws of physics tell us that a colder product will hold onto dissolved gas more readily than a room temperature one. Especially if your dessert wine is a white or a rosé, I’d guess you’d plan to serve it at least slightly chilled (54–56 °F/12–13 °C) if not all the way down to refrigerator temperature (37 °F/3 °C). Why not really “chilled,” like 50–52 °F (10–11 °C)? Try a range to find what best suits your wine, but I like to never serve dessert wines too cold, even whites. This is because if you chill it too much, delicate aromas are repressed and all you experience is a big whack of sugar on the palate with none of the retro-nasal nuance that should come with a well-made dessert wine.
Assuming you’ll be serving it somewhere between 52–56 °F (11–13 °C), try 7–10 PSI, which, if you think in beer terms, is about the fizz level of where a stout or a porter would be. I would start there first before going higher, which you certainly can. Carbonation works best when you give the gas some time to dissolve into the liquid. Ideally, you’d have a temperature-controlled place where this can happen. At my house, we’ve converted an older “garage fridge” into the “beer fridge” where we keep our kegs and sparkling water. In fact, while I was pregnant with my two sons, my husband made me my own kegs of ginger-apple sparkling water, which was essentially well-diluted apple juice flavored with fresh ginger. We found that if we chilled it to 45 °F (7 °C) then carbonated it and waited a few days, the gas would integrate a lot better and the bubbles would stay nicely in the glass. I’m not sure of your particular situation but having a fizz-friendly setup like that certainly makes it easy to dial in your temperatures and carbon dioxide levels.
Can you make a sparkling sweet wine with naturally fermented carbon dioxide? For the other readers out there who might be wanting to experiment, sweet sparkling wine is made by a) backsweetening during the “dosage” process in the méthode champenoise or charmat traditional sparkling process, b) capturing residual carbon dioxide gas during an arrested fermentation, or c) injected into an existing sweet wine, as seems to be the case here. Doing naturally sparkling wine right can take quite a bit of specialized equipment and know-how, but it certainly can be achievable on a small scale. But it is a challenge none-the-less for many hobby winemakers. If you’re interested in making naturally sparkling wine, be sure to check out the article starting on page 24 on making sparkling wines via the pét-nat method.