Place coconut flakes in food processor and pulse until finely chopped with some chunks of coconut still visible; do not process to a powdered consistency. Transfer to a bowl and mix with 1/4 cup of powdered sugar. Set aside.

Process coconut cookies in batches in the food processor until they become fine crumbs. When all cookies have been processed, measure 5 cups crumbs and pour them into a large mixing bowl. Discard any additional crumbs or reserve for another use.

Add remaining 1 cup powdered sugar, cocoa powder, cinnamon, and nutmeg to the mixing bowl and mix with cookie crumbs until well combined.

Mix together rum and sweetened condensed milk in a small measuring cup; pour slowly into the cookie crumb mixture, stirring to combine. Mix everything until well blended.

Use your hands to form dough into about thirty-six 1 1/4-inch balls. Roll each ball in the reserved coconut-sugar mixture until well coated, lightly pressing in the coconut topping. Store in an airtight container in the refrigerator until ready to serve or gift.