Eat a reasonable amount of protein. You probably don’t need to increase it, certainly not drastically. What we recommend is to keep carb intake under 20 g/day. Replace the calories no longer coming from carbohydrate with calories from fat. This is less than you probably think, since 1200 calories can come from either 300 g of carbohydrate or 133.3 g of fat.
The short version of all this is to restrict carbohydrate, eat a decent amount of protein, and add fat to satisfy your hunger.
Gluconeogenesis, the process by which the liver turns amino acids into glucose, is not, strictly speaking, driven by supply. The notion that “excess protein automatically turns to cake” was someone’s best guess at the time, but it has been shown by more-recent research to be unfounded. There are nuances, but broadly speaking, gluconeogenesis is more driven by demand than anything else. The body normally maintains no more than 5 g of glucose in the blood stream at any one time.
The protein we eat is needed mostly for structural purposes (muscle growth, tissue repair, and so forth), but amino acids are also the body’s only useable source of nitrogen for making nitric oxide, which is important for keeping blood pressure low.