Your ancestor served in what was one of the most heterogeneous military formations of the Napoleonic Wars, the Grande Armée of 1812. Even though there were considerable French presence in Napoleon's invasion force, it also included a considerable number of troops from areas under French control, ranging from Poland, the Italian peninsula, the Low Countries, and Spain. Most of these areas were either satellite states of the French Empire or areas directly annexed by France. Whether French-ruled or ruled via their proxies, one of the demands of Napoleon was warm bodies for his military ventures and these states instituted forms of conscription. The German states, organized into an entity called the Rheinbund (Confederation of the Rhine, collectively provided the largest contingent for Napoleon's invasion, ranging from 100-170000 men (depending on if one counts the Prussian and Austrian contingents as German).

So your family lore about a poor farmer freezing to death has a degree of validity. As you can see from the linked map, Southeast Germany in this period was either a Rheinbund state or directly annexed by France. Napoleonic conscription typically fell on those on the lower end of the socioeconomic scale, especially since it was a common practice for those men better off to pay for a substitute. Even without knowing more details, the smart money is that your ancestor was conscripted and served in one of the many non-French contingents serving under Napoleon.

If you're curious, the book Diary of a Napoleonic Footsoldier is an excellent primary source written by a German from this period and might give you some insight into what your great (5X) grandfather experienced.