Turn on a multi-functional pressure cooker (such as Instant Pot) and select Saute function. Add turkey and onion; cook and stir until turkey is browned and crumbly and onion is translucent, 5 to 7 minutes.
Add cabbage, diced tomatoes, chicken broth, water, rice, Worcestershire sauce, 1 teaspoon basil, salt, and pepper. Stir occasionally until mixture starts to boil, 5 to 7 minutes. Make sure the mixture does not exceed the max fill line.
Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 5 minutes. Allow 10 to 15 minutes for pressure to build.
While the cabbage-rice mixture is cooking, preheat the oven to 350 degrees F (175 degrees C).
Release pressure using the natural-release method according to manufacturer's instructions, about 8 minutes. Then use the quick-release method by gently pushing the steam release knob towards the venting position without pushing it fully into position; release remaining pressure in short bursts.
Remove the Instant Pot lid and stir in 1 cup of Cheddar cheese.
Combine tomato sauce, sugar, and remaining basil in a bowl. Pour 1/2 of the sauce into a 9x13-inch baking dish. Spoon cabbage-rice mixture over the sauce, then spoon remaining sauce over top. Sprinkle with remaining Cheddar.
Bake in the preheated oven until heated through, 25 to 30 minutes.