Add everything listed under “rum-infused fruit” to an air-tight container. Let this mixture soak overnight or up to 1 week (this will help plump up the fruit).
To prepare the caramel sauce, add everything listed under “caramel sauce” to a saucepan over medium to high heat. Whisk for about 5 minutes - keep a close eye on it - it'll bubble up.
Add the rum-soaked dried fruit mixture to the saucepan and let it cook for another 5 minutes - make sure to continue to stir to prevent burning.
Turn off the stove and allow the mixture to cool on the stove.
In a bowl, add the almond flour, coconut flour, tapioca flour, and baking soda and mix well. Add the eggs and mix until well combined. Fold in the fruit mixture until well combined, then spoon the batter into a greased 7-inch cake pan with a removable bottom (I also like to add a circle of parchment paper to the bottom of the pan for easy removal).
Bake for 350F for 1 hour - check on it around the halfway mark and cover with foil to prevent the top from getting too brown.