Preheat the oven to 350°F. Line an 8" square pan with parchment or a reusable liner and lightly grease. , In a small saucepan or in a microwave-safe bowl, combine the butter and fresh ginger. Heat over medium-low, or microwave in 30-second increments on half-power, until just melted. , Transfer the butter mixture to a large bowl and whisk in the brown sugar, ginger syrup, ground ginger, and salt until smooth and well-combined. Add the egg and whisk until shiny and emulsified. , Whisk in the baking powder, then whisk in the flour and diced ginger until just combined. Transfer the batter to the prepared pan and spread it to reach all of the corners. , Bake the ginger bars for 15 to 20 minutes, until set in the center, then let cool slightly in the pan before transferring to a wire rack to cool completely. , While the bars bake, make the frosting. Working with an electric mixer in a medium bowl or with a stand mixer fitted with the flat beater, beat the butter, cream cheese, and salt until very smooth. Add the confectioners’ sugar, lime zest and juice, and rum; beat until smooth and spreadable. Spread the frosting onto the cooled bars, then sprinkle evenly with the mini diced ginger. Slice and serve at room temperature. , Store leftover ginger bars well-wrapped at room temperature for up to 3 days. The frosting will taste more like rum as it sits.