Sear pork shoulder: Toss pork with 1 teaspoon of the salt. Select SAUTÉ setting on a programmable pressure multicooker (such as Instant Pot). (Times, instructions, and settings may vary according to cooker brand or model.) Select HIGH temperature setting, and allow to preheat 2 to 3 minutes. Add oil and one-third of pork to cooker. Cook until browned on one side, about 2 minutes; transfer to a plate. Repeat, in 2 more batches, with remaining pork.

Add remaining ingredients to cooker: Add onion to cooker and cook, stirring often, until softened, about 3 minutes. Stir in tomato paste, chili powder, garlic, cumin, and smoked paprika; cook, stirring constantly, until toasted and fragrant, about 1 minute. Add chicken broth and barbecue sauce, stirring to thoroughly combine. Stir in pork and beans.

Cook chili: Cover cooker with lid, and lock in place. Turn steam release handle to SEALING position. Select MANUAL/PRESSURE COOK setting. Select HIGH pressure for 50 minutes. (It will take 10 to 15 minutes for cooker to come up to pressure before cooking begins.)

Make sour cream mixture: While chili cooks, whisk together sour cream, vinegar, and remaining ¼ teaspoon salt in a medium bowl. Add slaw mix and ¼ cup of the scallions; toss to coat.

Vent cooker: When cooking has finished, carefully turn steam release handle to VENTING position, and let steam fully escape (float valve will drop). This will take 2 to 3 minutes. Remove lid from cooker.

Shred pork: Press each pork chunk against side of cooker to shred; stir to combine.

Serve: Top each serving with slaw, cheese, and remaining scallions. Serve with cornbread on the side.