Place coconut in a food processor and pulse until coconut is a fine powder. Add powdered sugar and continue to pulse until well mixed. Butter the bottom of an 11x17 sheet pan and top with parchment paper. Butter parchment paper and dust with powdered sugar mixture.
In a stand mixer, combine gelatin and 1/3 cup water. Let stand.
In a medium sauce pan with a heavy bottom combine sugar, remaining water, corn syrup, and salt. Slowly bring to a boil over medium heat. Continue to cook until mixture reaches the soft ball stage on a candy thermometer (about 240 degrees).
Turn on stand mixer and pour in sugar mixture in a slow steady stream. Leave mixer running on high speed for 15 minutes until mixture is light and fluffy.
Blend rum into marshmallow mixture and pour marshmallow onto prepared pan and smooth out to the corners. Evenly coat the top with 1/2 cup powdered sugar mixture. Let sit uncovered for 12 hours. Cover powdered sugar mixture and reserve.
Turn out onto a cutting board dusted with powdered sugar mixture. Butter and dust a cookie cutter with powdered sugar mixture. Cut marshmallows dip each cut edge into powdered sugar. Store in an airtight container for up to 2 weeks.