I don’t like the term “clean”, but I do recommend whole foods if you have a hard time controlling yourself. However, you can certainly lose fat while still eating pizza and desserts. It’s just a matter of how much.
If I were you, I’d drop your training frequency to 3 or 4 times per week and backload after every workout. The truth is, unless you are very advanced, you probably do not have the ability to fatigue yourself 5 times per week. CBL works best with highly fatiguing workouts, and you will likely get more muscle growth and strength gains by lowering your frequency.