Combine achiote paste, white onion, garlic, orange juice, lemon juice, 1/4 cup vinegar, 2 tablespoons salt, 1 tablespoon black pepper, and 1 tablespoon Mexican oregano in a blender; blend marinade until smooth.

Rub pork with salt and pepper and place in a pressure cooker. Add lard and pour marinade over pork. Add 2 cups water. Close cooker securely and place pressure regulator over vent according to manufacturer's instructions. Heat to High heat until steam escapes in a steady flow and makes a whistling sound, about 5 minutes. Reduce heat to medium-low or low. Cook for 45 minutes from the start of the whistling sound. Let pressure release naturally according to manufacturer's instructions, 5 to 10 minutes. Unlock lid and transfer pork to a large platter.

Prepare the habanero sauce while the meat is cooking. Combine red onion and habanero chiles in a gravy boat. Add 1/2 cup lemon juice, 1/4 cup white vinegar, and warm water. Season with salt, pepper, and dried oregano and mix until everything is well combined.

Shred meat with 2 forks and return to the pressure cooker with the cooking juices. Bring to a boil over medium-high heat and cook uncovered until cooking juices have reduced, about 30 minutes. Serve with habanero sauce.