Turn Instant Pot® to Normal Sauté. Add the oil and let heat for 1–2 minutes. Cut roast into 2 large pieces and add to the pot; let cook for about 3 minutes on each side, until browned. Press Cancel.
In a small bowl, whisk together the salt, root beer, and 1/2 cup of barbecue sauce; pour evenly over the roast. Place lid on pot and lock into place to seal. Pressure Cook or Manual on High Pressure for 45 minutes. Let sit for 10 minutes. Use Quick Pressure Release. Remove meat from the pot and place on a large cutting board; let cool to warm.
Turn pot to High Sauté, and simmer the juices for about 10–12 minutes, whisking frequently until thickened and reduced to about 1 cup of sauce. Stir remaining 1/2 cup barbecue sauce into the pot. Shred the meat and place back into the pot; tossing to coat with the sauce. Serve on buns or as a main dish.