Combine the dried dates, figs and cranberries in a medium microwave-safe bowl; add 2 tablespoons rum and the butter. Microwave in 1-minute intervals, stirring, until the dried fruit is softened and the butter is melted, about 3 minutes. Set aside.
Toast the pecans in a small dry skillet over medium heat, stirring occasionally, 3 to 4 minutes. Transfer to a cutting board and let cool, then coarsely chop.
Combine the graham crackers, crystallized ginger, candied lemon peel, molasses, cinnamon, allspice and salt in a food processor. Add the remaining 1 tablespoon rum, the pecans and the dried-fruit mixture (with any liquid from the bowl). Pulse until the mixture is finely chopped and holds together when squeezed. (If the mixture is dry, add a little water just until it holds together.)
Roll tablespoonfuls of the mixture into balls and arrange on a baking sheet. Refrigerate until firm, at least 2 hours or overnight.
Put the sanding sugar on a plate. Roll the balls in the sugar to coat.