Arrange a rack in the middle of the oven and heat to 425°F. Line a baking sheet with aluminum foil. Toast the panko while the oven is heating.
Heat the oil or butter in a large frying pan over medium heat until shimmering or melted. Add the panko and 1/4 teaspoon of the salt and toast, stirring frequently, until evenly light golden-brown, about 5 minutes. Transfer to a pie dish or large shallow plate and let cool completely. (This can be done up to 2 days ahead and stored at room temperature in an airtight container.)
Mix 1/2 cup of the mayonnaise, relish, capers, and mustard together in a small bowl to make the tartar sauce, set aside.
Cut the fish fillets as needed into pieces that will fit in the buns. Season the fish with salt and pepper. Brush both sides of the pieces with the remaining 1/3 cup of mayonnaise, then dredge in the toasted panko to completely coat, using your hands to press the crumbs into the fish as needed. Place on the baking sheet.
Bake until the fish is golden-brown and flaky, 12 to 14 minutes, depending on the thickness of the fish. Remove the pan from the oven. Lightly toast the rolls cut-side up directly on the oven rack, about 3 minutes.
Spread the tartar sauce on the cut sides of both the tops and bottoms of the rolls. Divide the lettuce over the bottom halves of the rolls, then top with the fish. Divide the tomatoes over the fish, if using, then close each sandwich with the top halves of the rolls. Cut each sandwich in half and serve.