To make the popsicles, combine the reserved ginger beer, 1/4 cup ginger simple syrup, dark rum, and lime juice, and pour into popsicle molds, leaving about 1/2-inch of space at the top to allow for lime addition and expansion as it freezes. Push a slice of lime into each mold, set sticks in place, and freeze. Since they're boozy, popsicles will cure best when frozen overnight.