Preheat oven to 400 degrees F (200 degrees C). Grease a 9x13-inch baking dish.
Combine lemon juice, sugar, butter, and rum in a saucepan. Bring to a simmer over low heat, stirring occasionally, until sugar is dissolved and sauce thickens slightly, about 10 minutes.
Arrange bananas cut-side down in the baking dish. Pour enough sauce over bananas to cover, reserving any extra sauce for serving.
Bake in the preheated oven until bananas are soft and sticky, about 30 minutes.
Bring water and brown rice to a boil in another saucepan. Reduce heat to medium-low, cover, and simmer until rice is tender and liquid has been absorbed, 45 to 50 minutes.
Combine black beans and diced tomatoes in a skillet over low heat; heat until warmed through, about 10 minutes.
Heat oil in a large skillet over medium heat. Add sausage links; cook, flipping often, until no longer pink in the center, about 10 minutes. Slice sausage into small pieces.
Scoop rice into a large bowl. Pour black beans on top. Sprinkle dill and basil on top. Add sausage; mix gently to combine. Serve bananas alongside.
