Bring broth, onion, 1 1/2 cups tomatoes, potatoes, basil sprigs, red-pepper flakes, and 2 teaspoons salt to a boil in a large, deep, straight-sided skillet with a tight-fitting lid. Reduce heat and simmer, uncovered, until potatoes are crisp-tender, 6 to 8 minutes. Season cod with salt and pepper, add to broth mixture, and cover. Simmer until fish is opaque throughout and just cooked through, about 7 minutes.
Remove and discard basil sprigs. Add sugar snaps, remaining 1/2 cup tomatoes, and lemon juice to skillet, gently stirring to combine; cook just until warmed through.
Divide fish, vegetables, and broth among 4 bowls. Garnish with basil leaves, drizzle with oil, and serve with lemon wedges for squeezing.