Place the chopped bacon in the Instant Pot and turn on the sauté function. Cook the bacon, stirring frequently, until crisp. Remove to a paper towel lined plate, leaving as much of the bacon fat in the pot as possible.
Add the onion, celery, and garlic and season with salt and pepper. Cook until softened, about 4 minutes. Turn off the sauté function.
Add the chicken broth to deglaze the pan, then add the cauliflower. Lock the lid onto the pot and make sure the vent is sealed. Set the manual function on high for 5 minutes (it will take 5 to 10 minutes to come up to pressure).
When the timer goes off, let the pressure release naturally for 10 minutes before opening the vent. Remove the lid and stir in the sour cream and 1 cup of the shredded cheese. Blend with an immersion blender until smooth or transfer to a full size blender or food processor.
Serve the soup topped with the remaining cheese, the bacon, and the green onion.