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Perhaps 20 years since I last made this cocktail so I'll remake and look at the recipe plus add booze and sweetness ratings (the fact these are missing shows a very old cocktail). That said, "Dolce" does mean 'sweet'.
I think it can also mean “good” in the deep sense, La Dolce Vida = the Good/Sweet life. I take the “dolce” in the name to be praise for lost Havana, more than the drink.