In a large skillet over medium heat, heat 1 Tbsp. oil; swirl to coat pan. Add celery, bell pepper, and scallions and cook, stirring frequently, until slightly softened, 2 to 3 minutes. Add garlic; season with salt and cook, stirring, until fragrant, about 1 minute more. Transfer celery mixture to a large bowl and let cool slightly.
To bowl with celery mixture, add egg, mayonnaise, mustard, parsley, Cajun seasoning, Worcestershire, lemon pepper seasoning, and black pepper and whisk to combine. Using a spatula, gently fold in crabmeat and crushed crackers without breaking up meat too much. Using a ¼-cup measuring cup, portion crabmeat mixture into 10 patties.
Wipe skillet clean. In same skillet over medium heat, heat 2 Tbsp. oil until shimmering. Add 5 patties and cook, turning halfway through, until both sides are deep golden brown, 3 to 4 minutes. Transfer to a plate. Repeat with remaining 2 Tbsp. oil and 5 patties.
Spread remoulade on cut sides of buns. Top with crab cake slider, a slice of tomato, and a lettuce leaf. Close sliders to serve.
