In a mixing bowl, beat butter and cream cheese with an electric mixer until combined, about 30 seconds; beat in the confectioners' sugar until the mixture is light and fluffy, about 2 minutes. In a bowl, whisk the flour with salt, and stir the flour mixture into the butter mixture to make an even, smooth dough. Cover the dough, and chill in refrigerator for at least 3 hours or overnight.
Preheat oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper.
Roll the dough on a well-floured work surface to a thickness of about 1/8 inch. Cut the dough into rounds with a 3-inch round cookie or biscuit cutter. Place the cookies onto the prepared baking sheet. Mix the mincemeat and rum in a bowl, and spoon about 1 teaspoon of the filling into the middle of a cookie. Bring together 2 adjacent edges of the cookie into a triangle shape, and pinch the cookie closed to partially cover the filling and make a cornucopia or bell shape.
Bake in the preheated oven until the cookies are light golden brown, 10 to 12 minutes. Allow to cool on the baking sheet for about 5 minutes before removing to wire racks to finish cooling.