Turn on a multi-functional pressure cooker (such as Instant Pot(R)) and select Saute function. Heat 1 tablespoon avocado oil and tilt the pot to distribute the oil evenly. Once oil shimmers, add 1/2 the beef and cook until browned, about 5 minutes. Transfer browned beef to a bowl and repeat with remaining beef.
Heat another tablespoon oil in the Instant Pot(R); when oil shimmers, add mushrooms. Cook mushrooms, stirring occasionally, until slightly brown, 4 to 5 minutes. Add mushrooms to the bowl with the beef. Add 1 tablespoon avocado oil to the pot and cook carrots, onions, and celery 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 barley, tomato paste, bay leaves, thyme, bouillon base, red pepper flakes, and beef broth. Stir until well combined.
Close and lock the lid. Select high pressure according to manufacturer's instructions; set 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.
