TL;DNR: You would need to change Blender's C source code to make the information available.
Obtaining that information at run time would require being able to register an callback handler for baking. These are sometimes called callbacks, but the blender Python manual calls them handlers.
There are no callbacks for baking, despite this request on RightClick. Someone even tried to file this bug report.
You can find the mostly complete list of available app callback handlers in the manual.
The other way that can sometimes be used is Blender's version of Message Bus but that only works if the change in the displayed percentage were available through what is called the "RNA" system. Baking does not expose its progress percentage this way.