Cut the pork roast into 5-6 pieces, making them as equal as possible. Season generously with barbecue seasoning.
Turn the Instant Pot onto SAUTE. When the pot reads "HOT," add 1 tablespoon olive oil to the inner pot. Add 2-4 pieces of pork to the pot in a single layer and sear on all sides. Repeat with remaining pork.
Deglaze the pot with the beef broth, using a wooden spoon or spatula to get any remaining browned bits off the bottom. Pour in the Dr Pepper, add the trivet to the pot. Lay the seared pork pieces on the rack. Pour 2 cups of the ketchup over the top.
Place and lock the lid on the pot. Make the sure the valve is turned to "SEAL" and set pot for 75 minutes at HIGH pressure.
When the pot finishes cooking, natural release for 15 minutes, then quick release the rest of the pressure. Use tongs to remove the pieces of pork to a large bowl and shred.
Use a gravy separate and carefully pour the drippings through the strainer. Do this until you have 6 cups of drippings (may take more than once, depending on the size of your separator.
Press CANCEL/OFF, then SAUTE. Pour the drippings back into the inner pot and whisk in the remaining cup of ketchup, as well as the liquid smoke. Bring the mixture up to a simmer and cook until reduced and thickened into a sauce. Pour over the shredded pork and stir to coat.