Hello Noah,
From your latest remarks it looks like you have a good handle on your watch. I have to agree with you, why would they have 2 different versions ( 12 Hour versus 24 Hours).
Beats me and does not make sense. The 24 Hour hand needs additional gearing of the hour drive. I doubt that there is an internal adjustment but you can check.
A battery change would not cause a "switch" from one mode to another. It does not effect the movement.
It is a great looking watch and it would be nice if the intended function could be realized. One last suggestion : maybe the HEUER technical department has an answer, it's worth a try. They are usually quite helpful when there is questions about their products.
Let me know if and when you find out more about this unique situation. Good luck and take care. Heinz.