In a medium bowl, whisk together the sour cream, vinegar, ½ teaspoon salt, and ¼ teaspoon pepper. Add the cabbage and carrots and toss well.
Season the tilapia with ½ teaspoon salt and ¼ teaspoon pepper. Place the flour in a medium bowl, the eggs in a second medium bowl, and the panko in a third medium bowl. Coat the tilapia with the flour (tapping off any excess), dip in the eggs (shaking off any excess), then coat with the panko (pressing gently to help it adhere).
Heat the oil in a large nonstick skillet over medium-high heat. Cook the tilapia in batches until cooked through, 2 to 3 minutes per side, adding more oil to the pan if necessary. Serve with ketchup and coleslaw.