Place crumbled brownies in a bowl. Pour rum over brownies, let soak for 1 minute; stir until mixture forms a paste-like consistency.
Using a 1-inch sorbet scoop or melon baller, scoop out balls of chocolate-rum mixture and transfer each to the plate lined with plastic wrap. Place the plate with chocolate balls in freezer for 15 to 20 minutes, or until firm enough to roll. Remove from freezer, roll each into a smooth ball; return plate to freezer.
Melt 3 ounces chocolate in the top of a double boiler over simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching. Remove from heat. Add remaining 1 ounce chocolate; stir until chocolate is melted and combined.
Remove chocolate balls from freezer; place one ball in melted chocolate and gently move around with a fork to coat completely. Transfer to baking sheet lined with a silicone baking mat. Repeat for remaining chocolate balls.
Dust chocolate rum balls with cocoa powder and serve.