Turn on a multi-functional pressure cooker (such as Instant Pot) and select Sauté function. Heat 1 tablespoon avocado oil and tilt the pot to distribute oil evenly. Once oil shimmers, add 1/2 of the beef and cook in hot oil until browned, about 5 minutes. Transfer browned beef to a bowl and repeat with remaining beef.

Heat another 1 tablespoon avocado oil in the pot; when oil shimmers, add mushrooms. Cook mushrooms in hot oil, stirring occasionally, until slightly brown, 4 to 5 minutes. Transfer mushrooms to the bowl with beef.

Add remaining 1 tablespoon avocado oil to the pot. Cook and stir carrots, onions, and celery in hot oil until onions begin to soften, about 5 minutes. Add garlic and cook until just fragrant, about 1 minute. Pour in wine to deglaze the pot, scraping up any browned bits, and cook until wine has almost evaporated, 3 to 5 minutes. Return browned beef and mushrooms to the pot. Stir in beef broth, barley, tomato paste, bay leaves, thyme, bouillon base, and red pepper flakes until well combined.

Close and lock the lid. Select high pressure according to manufacturer's instructions; set the timer for 30 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure using the natural-release method according to manufacturer's instructions for 15 minutes. Release remaining pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Remove and discard bay leaves. Taste and adjust seasoning with salt.