Turn on a multifunctional pressure cooker, such as Instant Pot, and select the Sauté function. Add olive oil. Add bacon and pork neck bones to hot oil. Cook until browned on both sides, 5 to 7 minutes. Remove meat and set aside. Add bell pepper, celery, and onion; cook and stir for 1 to 2 minutes. Season with white pepper, thyme, garlic powder, oregano, cayenne pepper, Cajun seasoning, and black pepper. Mix in garlic and continue cooking until onion is translucent, 2 to 3 minutes.

Mix kidney beans, chicken broth, salt, and bay leaves into vegetables in the pot; stir well to combine. Cancel Sauté mode and return meat to the pot. Close and lock the lid. Set vent to Sealing. Select High pressure, according to the manufacturer's instructions. Set timer for 60 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure using the natural-release method, according to the manufacturer's instructions, about 10 minutes. Move the valve to the venting position and finish releasing pressure, about 5 minutes. Unlock and remove the lid.

Remove pork bones from the pot. Use a fork to separate meat from bones and return meat back to the pot. Mix well. Serve with white rice.