Using the highest sauté or browning function on the Instant Pot, heat the vegetable oil until shimmering. Add the chopped onion and cook until softened, about 3 to 5 minutes. Stir in the garlic, ginger, and red curry paste, cooking until fragrant, about 30 seconds.
Add the broth, half the coconut milk, fish sauce, and rinsed wild rice. Scrape up any browned bits from the bottom to prevent burning.
Secure the Instant Pot lid and close the pressure valve. Set to high-pressure cook for 30 minutes. Once finished, turn off the cooker and use the quick-release method to release the pressure. Carefully remove the lid, allowing steam to escape away from you.
Stir in the remaining coconut milk, lime juice, and sugar. Add the mushrooms and cook using the highest sauté or browning function until tender, about 3 to 5 minutes.
Turn off the Instant Pot, stir in the chopped cilantro, and adjust seasoning with extra fish sauce, salt, and pepper to taste.
Ladle the soup into bowls and serve with lime wedges for added freshness.