Whisk sweetened condensed milk and eggs together in a bowl; add milk, 1 1/2 teaspoons syrup from canned pineapple, and rum and stir well.
Pour remaining syrup from canned pineapple into a saucepan and cook over medium heat until bubbling and syrup darkens, 2 to 3 minutes. Remove saucepan from heat and pour syrup into a custard dish and tilt to completely cover the bottom. Arrange pineapple slices in the syrup and pour milk mixture over pineapple.
Bake in the preheated oven until a toothpick inserted in the center comes out clean, about 1 hour. Cool to room temperature and refrigerate until chilled, at least 1 hour.