K31s have a fully supported chamber. That warning was for 1911 and older models. Civilian K31s were chambered in 308, 30-06, and 300WM. They are plenty strong and safe. Commercial hunting ammo is available from Privi and Hornady.
A lot of the fears propagated about straight pull rifles really don't apply to the K31 and probably come from horror stories concerning the Ross rifle. If you reload the 7.5 Swiss in a K31 is essentially the same performance as a 308, perhaps better with heavier bullets. The real issue with rechambering these rifles is the helical locking lugs. Typically a new barrel is attached to a stub from the existing barrel. Go to Swissrifles.com and you should get some good advice.