Beat eggs together in a large bowl with an electric mixer on medium speed until foamy, 2 to 3 minutes. Beat in 1 1/2 cups sugar, milk, cream, rum, vanilla extract, and salt until thoroughly combined. Gently stir in bread cubes to coat; soak until egg mixture has been absorbed, about 1 hour, stirring occasionally.

Dissolve remaining 1 cup sugar in 1/2 cup water and vinegar in a heavy saucepan over high heat; cook without stirring until dark amber in color, about 10 minutes. Remove from heat; cool until stops bubbling. Gradually stir remaining 1/4 cup water into caramel sauce, about 1 tablespoon at a time, stirring until water is incorporated.

Pour warm caramel sauce into a 9x9-inch baking pan; swirl the pan to coat the bottom and sides. Let caramel sauce set, about 15 minutes. Coat inside of the baking dish and caramel layer with butter. Arrange soaked bread cubes and any remaining liquid on caramel layer; gently press bread cubes together. Cover pan with parchment paper.

Preheat the oven to 350 degrees F (175 degrees C).

Place baking pan into a large roasting pan; pour enough boiling water into the roasting pan to reach halfway up the sides of the baking pan.

Bake in the preheated oven until a knife inserted into center comes out clean, about 1 1/4 hours. Let stand for 5 to 10 minutes. Run a knife around the outside of the pan; invert pudding and caramel sauce onto a serving platter.