I copied this info off the Swiss Rifle site pertaining directly about the Schmidt-Rubin K31.
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The new carbine also had several other new features. The barrel was intended to be free floating. The action itself only connected to the stock by two screws, one attaching to the chamber, with the second attaching to the tang. This allowed the Swiss to eliminate the aluminum barrel collar used in the Schmidt-Rubin series. The trigger was redesigned, as were the magazine and rear sight.
The end result of these modifications produced a rifle that was more accurate, yet cheaper than the K11, it was replacing. In 1932 it was estimated that the cost of a K11 would be 169 Swiss Francs. Conversely, the new carbine had an estimated cost of 151 Swiss Francs.
On March 17, 1932, the Swiss Upper Parliament accepted the new carbine into service and designated it the Model 1931.
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So according to this article, you should be able to restock your K31 with a sporter stock with no problem.