Melt the butter in a small saucepan over high heat, add the brown sugar and rum and cook, whisking until the sugar has melted and the mixture is smooth. Whisk in the heavy cream and cook until heated through and slightly thickened. Transfer the mixture to a bowl, keep warm.
Heat the grill to high.
Brush the pineapple on both sides with the oil and place on the grill. Grill until golden brown and caramelized on both sides, about 6 minutes. Remove from the grill, stack the slices and cut into chunks. While the pineapple is grilling, place the pound cake on the grill until lightly golden brown on both sides, about 20 seconds on each side. Remove the pineapple from the grill and, when cool enough to handle, either slice or cut into chunks.
Top each piece of pound cake with some of the pineapple and drizzle with the caramel-rum sauce. Top with more pound cake, vanilla ice cream and a maraschino cherry, if desired.