Cook bacon in Instant Pot on Sauté setting, stirring occasionally, until browned and most of the fat is released, 6–8 minutes. (If using a stovetop pressure cooker, cook bacon over medium-high heat.)
Meanwhile, cut onion into quarters; set one quarter aside. Coarsely chop remaining onion. Add chopped onion, carrots, and garlic to Instant Pot, season with a big pinch of salt, and cook on Sauté, stirring occasionally, until vegetables are softened but haven’t taken on color, 6–8 minutes.
Add chiles along with their liquid, adobo sauce, oregano, and cumin to Instant Pot and cook, stirring, until fragrant, about 1 minute. Add beans, 2 tsp. salt, and 2 quarts water. Lock lid and cook soup at high pressure 30 minutes. Release pressure manually. Taste soup and season with more salt if needed.
Meanwhile, thinly slice reserved onion and transfer to a small bowl. Add lime juice and remaining ½ tsp. salt. Let sit, tossing occasionally, until ready to use.
Using an immersion blender, purée soup in pot until smooth (or, working in batches, purée with a regular blender).
Divide soup among bowls. Top with sour cream, tortilla chips, cilantro, cheese, and drained pickled onions. Serve with lime wedges for squeezing over.