Combine lemongrass, chile, bay leaf, coriander seeds, mustard seeds, sugar, 1/4 cup salt, vinegar, and water in a saucepan. Bring to a boil, stirring once. Reduce heat, and simmer for 10 minutes. Transfer to a measuring cup; let cool.
Preheat grill to medium-high. (If you are using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 3 to 4 seconds.) Brush eggplant slices with oil, and grill, flipping once, until cooked through and blackened in places, about 2 minutes per side. Let cool slightly.
Place snapper into a skillet, and cover with water by 1 inch. Sprinkle with 2 tablespoons salt. Bring to a boil. Remove from heat, and cover with a round of parchment or a lid; let stand until snapper is opaque and cooked through, about 10 minutes. Using a slotted spatula, carefully transfer snapper to a plate. Cover with parchment to keep it from becoming dry. Discard poaching liquid. When snapper is cool enough to handle, gently flake into 1-inch pieces.
Arrange grilled eggplant, poached snapper, and melon slices on a large platter. Drizzle with 1 cup lemongrass sauce. Cover platter with plastic wrap, and refrigerate, along with remaining lemongrass sauce, for at least 2 hours (or up to 4 hours). Serve cold with extra sauce on the side.