Turn on a multi-functional pressure cooker (such as Instant Pot), and select Saute function. Add butter and olive oil; heat until butter is melted. Add carrots, celery, and onion; cook until onion is nearly translucent, about 3 minutes. Add ham and cook for 1 minute.
Add potato; cook and stir to prevent anything from sticking to the bottom of the pot, for 2 minutes. Season with salt and pepper. Add 3 cups chicken broth, split peas, and bay leaves. Turn Saute function off.
Close and lock the lid and seal the valve. Select high pressure according to manufacturer's instructions; set timer for 15 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, 10 to 12 minutes. Release remaining pressure carefully using the quick-release method, 2 to 3 minutes. Unlock and remove the lid.
Unlock and remove the lid. Taste and adjust with salt and pepper as desired. If soup is too thick, select Saute function and add remaining 1 cup chicken broth. Stir until heated through and desired consistency, about 5 minutes. Adjust seasonings if necessary. Serve immediately with Cheddar cheese sprinkled over top.