Combine the flour, salt and baking powder in a medium mixing bowl and set aside.
In an electric mixer fitted with the paddle attachment, beat the butter at medium speed for about 1 minute. Gradually add the sugar and continue to beat until it is pale and fluffy, an additional 3 to 4 minutes. Add the egg and continue to beat 1 more minute. Stir in the rum and lime zest.
Add the flour mixture to the creamed butter all at once then stir with a wooden spoon or spatula for a couple of turns. Return the bowl to the stand and mix on the lowest speed until the flour is just incorporated, do not over-mix. Shape the dough into 2 equal discs, and cover each in plastic wrap. Refrigerate until very firm, at least 2 hours or overnight.
Preheat the oven to 350 degrees F. Line a large baking sheet with parchment paper or nonstick liner.
Place one portion of dough between 2 pieces of parchment or wax paper (keep remaining dough chilled) and roll out to about 1/2-inch thick. Repeat with remaining dough. Return the rolled discs to the refrigerator between layers of parchment and chill until firm again, 15 to 20 minutes.
Using a round cookie cutter (about 1 1/2-inch wide) or other desired shape, cut out the cookies and place on the lined baking sheet. Re-roll scraps once to cut more shapes. Using a small teaspoon or narrow end of a pestle, make a small indentation in the center of each cookie. Chill shaped cookies until firm, 5 minutes, while you prepare the guava filling.
In a small saucepan, melt guava preserves over low heat, 5 minutes, stirring constantly until smooth. Strain preserves into a bowl.
Place about 1/4 teaspoon of melted guava in the center of each cookie, do not overfill. Bake the cookies until the edges are lightly golden, though the tops will still be pale, 15 to 18 minutes. Using a spatula, carefully remove the cookies and lay out on a cooling rack.