Remove and discard turkey giblets and neck. Place turkey on a wire rack in a large roasting pan; pat dry. Whisk together pineapple juice and 1 tablespoon of the salt. Using a meat injector, inject 1/4 cup of the juice mixture into each breast; let stand at room temperature 1 hour.
Preheat oven to 400°F. Place 1 cup of the rum in a small saucepan over medium-high. Cook until reduced by half, about 7 minutes. Remove from heat. Whisk butter pieces into rum until melted.
Sprinkle turkey cavity with 1 1/2 teaspoons of the salt and 1/4 teaspoon of the pepper. Insert carrots, celery, onions, sage, and thyme sprigs into cavity. Scatter any remaining vegetables around turkey in roasting pan. Tie turkey legs together with kitchen twine. Brush rum-butter mixture generously over turkey. Pour any extra mixture into roasting pan. Sprinkle turkey evenly with 1 tablespoon salt and 3/4 teaspoon pepper. Pour 1 cup of the stock into roasting pan.
Bake turkey in preheated oven until deep golden brown, about 1 hour and 30 minutes, basting with pan drippings every 30 minutes. (Cover turkey with aluminum foil if it is getting too brown on top.) Reduce oven temperature to 350°F, and bake until a meat thermometer inserted in thickest portion registers 165°F, 45 minutes to 1 hour, basting with pan drippings every 30 minutes. Remove turkey from pan; remove and discard vegetable mixture from pan and inside turkey cavity. Let turkey stand 20 to 30 minutes before carving.
Pour pan drippings through a fine wire-mesh strainer over a large saucepan; discard solids. Let stand 5 minutes. Skim fat from top of drippings. Stir in 1 1/2 cups of the stock and remaining 1/2 cup rum, and bring to a boil over medium-high. Whisk flour into remaining 1/2 cup stock in a small bowl; slowly add to drippings mixture, whisking constantly. Cook, stirring often, until thick and bubbly, about 5 minutes. Remove from heat; stir in chopped thyme, vinegar, and remaining 1/2 teaspoon salt and 1/4 teaspoon pepper.