Heat 1 tablespoon oil in a multi-functional pressure cooker (such as Instant Pot®) set on the Saute function. Add beef mince; cook and stir until browned, about 5 minutes. Season with salt and pepper. Drain beef and transfer to a bowl.
Heat 1 teaspoon oil in the pot. Add onion, carrots, celery, red pepper and garlic. Cook until soft, about 5 minutes. Add chilli powder, cumin and crushed chilli flakes; cook until fragrant, 1 to 2 minutes.
Pour beef stock into the pot; scrape the browned bits of mince off the bottom of the pot. Stir in cooked beef and beans. Add diced tomatoes but do not stir. Close and lock the lid. Select Bean/Chilli function; set timer for 10 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, 10 to 40 minutes. Stir in lemon juice and tomato puree. Let sit for 10 minutes before serving.