Preheat oven to 350°F. Butter a 4-by-8-inch loaf pan, line bottom with parchment paper and butter parchment; dust with flour.
In a small bowl, stir together eggs, 2 Tbsp. cream and 1 Tbsp. rum. Sift flour, 1/2 cup sugar, baking powder and salt into a large bowl. Add butter and half of egg mixture and beat with an electric mixer until well-blended, 1 minute, scraping down sides of bowl as needed. Add remaining egg mixture and beat well. Fold in coconut. Scrape batter into pan and bake until cake is light golden and a toothpick inserted in center comes out clean, about 45 minutes. Let cool for 10 minutes, then unmold, remove parchment and cool completely on a wire rack.
Wash and dry loaf pan and line completely with plastic wrap. Using a serrated knife, cut cake horizontally into 3 equal layers. Brush 2 Tbsp. rum evenly over cut sides of cake. Place bottom third in loaf pan. In a bowl, beat remaining 6 Tbsp. cream with remaining 1 Tbsp. sugar until stiff peaks form. In a small bowl, stir together pineapple and remaining 1 Tbsp. rum. Fold pineapple mixture into cream.
Spread ice cream evenly over cake in pan. Top with middle cake layer, then spread with pineapple-cream mixture. Top with remaining cake layer. Cover with plastic wrap and freeze until firm, at least 8 hours or up to 2 days.
Remove from freezer 15 minutes before serving. Unmold and peel away plastic wrap. Cut into 1-inch slices with a hot knife, and serve.