For the catfish: Line a sheet pan with a rack and set aside. Stir together 1 tablespoon of the paprika and the salt in a small bowl. Sprinkle the mixture over the catfish filets and refrigerate for 30 minutes.
Meanwhile, stir together the remaining 3 tablespoons paprika, garlic powder, onion powder, black pepper, cayenne, cumin, oregano, and thyme in a small bowl. Set 4 teaspoons of the spice mix aside; the rest will keep, in an airtight container, for up to 1 year.
When the catfish is ready, season each filet on one side with 2 teaspoons of the spice mix. Lightly oil a large cast-iron skillet and heat over medium-high heat until almost smoking. Place the filets in the skillet seasoned-side down and cook until lightly blackened, 1 to 2 minutes. Flip the filets, add the butter and cook 1 to 2 minutes more, occasionally basting with the melted butter. Transfer the filets to the prepared sheet pan and set aside.
For the fried okra: Whisk the eggs in a large shallow dish and season with salt and pepper. Add the okra and toss to combine. Place the cornmeal and cayenne in another large shallow dish, season with salt and pepper and stir to combine. Transfer the okra to the dish of seasoned cornmeal, letting any excess egg drip off, and toss until completely coated. Transfer the okra into a fine-mesh colander or hand strainer and gently shake to remove any excess cornmeal.
Add 1 to 2 inches of oil to a large nonstick skillet and heat to 350 degrees F. Working in batches, fry the okra, gently turning with tongs, until golden brown and crispy, 3 to 4 minutes. Transfer to the sheet pan with the catfish. Repeat with the remaining okra.
Plate the blackened catfish and fried okra next to each other and garnish with the lemon wedges.