Line a 9x9 inch square baking pan with parchment paper.
Melt the butter in the microwave or over the stove. Combine the melted butter and GF graham cracker crumbs in a food processor or high speed blender and pulse until a wet sand consistency forms. Firmly press the crumbs into the prepared baking pan using the bottom of a cup. Place the pan in the freezer while doing the next step.
Heat the cream cheese in the microwave for about 30 seconds and mix until smooth and slightly runny. Transfer the ice cream to a large mixing bowl and add the melted cream cheese. The slight warmth from the cream cheese should soften the ice cream but if not, let it sit for a couple minutes. Using a sturdy spatula, break up the ice cream and mix in the cream cheese. Mix as well and quickly as you can to ensure the cream cheese is fully combined and the ice cream doesn't melt. Quickly remove the graham cracker crust from the freezer and spread the cream cheese ice cream on top. Immediately return to the freezer.
In a saucepan over the stove, combine all the berry layer ingredients, mix well, and turn the heat to medium. Let the mixture come to a boil and continue to cook until the berries are very soft and mash-able. Transfer the sauce to a blender and mix until smooth or leave it in the saucepan and blend with an immersion blender. Pour the blended sauce through a fine mesh sieve (if you don't want seeds) and into a metal bowl. Let the mixture cool at room temperature for 15-20 minutes (20 is safer).
Once the sauce has cooled, remove the ice cream bars from the freezer, quickly pour all the sauce on top and tip and rotate the pan to completely cover the surface of the ice cream with the berry mixture. Immediately return to the freezer to chill for at least 30 minutes. After 30 minutes, set a cutting board, sharp knife, and large container on the counter. Remove the bars from the freezer, quickly pull the edges of the parchment paper to remove the whole batch of bars from the pan, set it on the cutting board, slice into 16 equal sized bars (doesn't have to be that specific), and immediately eat or place them in the container and return to the freezer. Enjoy!