Pour 1/4 cup of the dark rum over the raisins in a small bowl, cover, and soak for at least 30 minutes at room temperature (or overnight in the refrigerator). Drain any excess rum and reserve it for the custard.
Preheat oven to 350°F (175°C) and generously butter the 9x13-inch baking dish. Spread the bread cubes in an even layer, then scatter the rum-soaked raisins over the top — if using fresh bread, toast cubes for 5–7 minutes first to dry them out.
Whisk the eggs in a large bowl until smooth, then add the whole milk, evaporated milk, and brown sugar, whisking until no lumps remain. Stir in the melted butter, remaining rum, vanilla, cinnamon, nutmeg, allspice, and salt until fully combined.
Pour the custard evenly over the bread cubes and gently press the bread down to coat every piece. Let the mixture rest for 15–20 minutes, pressing down any floating pieces occasionally, until the custard is fully absorbed.
Bake on the center rack for 50–60 minutes until the top is golden brown and a knife inserted in the center comes out mostly clean with just a few moist crumbs. If the top browns too quickly, cover loosely with foil and continue baking.
Melt butter in a small saucepan over medium heat, whisk in the brown sugar until dissolved, then carefully stream in the heavy cream and simmer for 3–4 minutes until slightly thickened. Remove from heat and stir in rum, vanilla, and a pinch of salt.
Let the pudding cool for at least 15–20 minutes before cutting into squares. Serve warm, drizzled with rum sauce and topped with vanilla ice cream or whipped cream if desired.