Combine water and beans in a multi-functional pressure cooker (such as an Instant Pot). Close and lock the lid. Select high pressure according to manufacturer's instructions; set the timer for 15 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure using the natural-release method according to manufacturer's instructions, about 20 minutes; quick-release remaining pressure according to manufacturer's directions. Unlock and remove the lid. Drain and rinse beans with cold water and set aside. Rinse and wipe out the Instant Pot insert and place back into the pressure cooker.

Turn on the Instant Pot and select the Sauté function. Heat olive oil until shimmering, 2 to 3 minutes. Add salt pork, bacon, and onion; cook until fat begins to render, 1 to 2 minutes. Pour in 1/2 cup water and scrape any browned bits off the bottom. Turn the Instant Pot off.

Whisk together molasses, ketchup, brown sugar, mustard, and remaining 1 cup water in a small bowl. Return cooked beans to the pot along with ketchup mixture. Gently stir to combine. Close and lock the lid. Select high pressure according to manufacturer's instructions; set the timer for 35 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure using the natural-release method according to manufacturer's instructions, about 20 minutes; quick-release remaining pressure according to manufacturer's directions. Unlock and remove the lid. Beans will thicken as they cool. Serve immediately or freeze portions for later.