Heat 2 tablespoons oil in a large skillet over medium-high heat. Cook potatoes undisturbed 4 minutes, then stir and cook 8 more minutes until golden and fork-tender.
Push potatoes aside, add remaining oil, fennel, and onion. Cook 5 to 6 minutes until softened.
Stir in garlic, smoked paprika, salt, and pepper; cook 1 minute. Combine and push to the edges of the pan.
Pat catfish dry, season with Cajun seasoning. Add butter to the cleared center and sear catfish 2 to 3 minutes per side until golden and opaque.
Gently fold the catfish into the hash. Remove from heat, top with fennel fronds, and serve with lemon wedges.