Place rabbit in large stockpot with enough water to cover. Add 5 garlic cloves, 5 bay leaves, celery, onion, carrot, and 2 tablespoons salt. Cook over high heat until boiling, then lower heat and leave gently bubbling until tender, about 1½ hours. Remove from heat and allow to cool in liquid. Strain and reserve liquid for later use. Debone rabbit meat and finely shred. Set meat aside.
Place tomatoes, chipotle peppers and reserve liquid sauce, whole cumin seeds, black peppercorns, whole cloves, remaining bay leaves, and garlic in blender and puree until smooth. Set aside.
Place cast-iron skillet over medium heat, add lard to skillet, and preheat 2 minutes. Add remaining onion and cook until translucent, about 3 minutes. Add puree and cook 4 minutes, stirring occasionally. Add meat and cook 2 minutes. Stir in oregano, remaining salt and black pepper and cook for 1 minute more. Carefully fold in potato. Transfer tinga to a bowl. Refrigerate tightly covered with plastic wrap.