You’re talking about keto-adapted autophagy, which is entirely possible and seems like the best option for you. It is however not fasting to eat anything. Strictly speaking: any amount of calories, no matter how small, disrupts the fasting process.
I think a lot of people get confused because it is possible to preserve the benefits of intermittent fasting—burning fat for fuel, controlled blood sugar levels, etc.— but the definition of being in a fasted state is not generally recommended to consume anything other than a few select beverages.