Combine the strawberries, sugar, and 1/4 cup of water in a large saucepan and cook uncovered over medium-low heat for 5 minutes. Add 2 cups of the raspberries and all the blueberries and cook, stirring occasionally, until the mixture reaches a simmer, and simmer for a minute. Off the heat, stir in the remaining raspberries and the framboise.
Slice the bread in 1/2-inch-thick slices and remove the crusts. In the bottom of a 7 1/2-inch round by 3-inch high souffle or baking dish, ladle about 1/2 cup of the cooked berry mixture. Arrange slices of bread in a pattern (this will become the top when it's unmolded) and then add more berry mixture to saturate. Continue adding bread, cutting it to fit the mold, and berries. Finish with bread and cooked berries, using all of the fruit and syrup.
Place a sheet of plastic wrap loosely over the pudding. Find a plate approximately the same diameter as the inside of the mold and place it on top. Weight the mold with a heavy can and refrigerate. Remove the weight after 6 to 8 hours. Cover the pudding with plastic wrap and refrigerate overnight.
Just before serving, run a knife around the outside of the pudding and unmold it upside down onto a serving plate. Serve in wedges with rum whipped cream.
Whip the cream in the bowl of an electric mixer fitted with the whisk attachment. When it starts to thicken, add the sugar, vanilla, and rum. Continue to whip until it forms stiff peaks. Serve cold.
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