What is the ruling on fasting for someone who suffers from a stomach illness that causes increased saliva secretion and compels them to drink something hot or eat a small piece of bread or a small biscuit to avoid severe vomiting and excruciating pain?

If not breaking the fast causes what you have mentioned, then it is permissible for you to break the fast, and you must make up the days you missed after recovery, due to the Almighty's saying: "And whoever is ill or on a journey - then an equal number of other days."