Select the high Sauté setting on the Instant Pot and heat 2 tablespoons of the butter. Add the yellow onion, ginger, garlic, and bell peppers and sauté until the onion turns translucent about 5 minutes. Add the cauliflower, potatoes, tomatoes, peas, and 2 teaspoons salt; pour in the water; and stir well with a wooden spoon, nudging loose any browned bits from the pot bottom.
Secure the lid and set the Pressure Release to Sealing. Press the Cancel button to reset the cooking program, then select the Pressure Cook or Manual setting and set the cooking time for 6 minutes at high pressure.
Perform a quick release by moving the Pressure Release to Venting. Open the pot. Select the normal Sauté setting and mash all the veggies with a potato masher. Stir in the tomato paste, paav bhaji masala, red chile powder, and turmeric, followed by 2 tablespoons butter (optional but highly recommended). Cover the pot with a glass lid (or other nonlocking lid that fits), as the bhaji will start to splatter, and cook until heated through, about 5 minutes. Taste and adjust the seasoning with salt, paav bhaji masala, and chile powder if needed. Press the Cancel button to turn off the Instant Pot. Sprinkle with the cilantro.
Just before serving, use the remaining 4 tablespoons butter to spread 1 teaspoon room-temperature butter on the cut side and over the top and bottom of each potato bun. Heat a griddle or heavy skillet over high heat. Add the buns and toast each side until golden brown, 2 to 3 minutes each.
Spoon the bhaji onto plates, sprinkle with the red onion, and serve with the toasted buns and lime wedges on the side.