James B. Stewart wrote an excellent article in Gun Digest 1973, entitled "Bergmann System Military Pistols". He notes,
"Sometime during 1912 or 1913 a very small quantity of the M1910 type were delivered to Greece for testing. The Greeks did not adopt this arm and the fate of the test pieces is unknown. Several examples in the proper serial number range have been found wit the inscription "L.Sk. 16." followed by another two-digit number stamped on the front grip strap. These may have been Greek test pieces or the marking may indicate some police usage. Several countries bought small quantities of the Bergmann-Bayard for issue to their police forces...".
The other Bergmann-Bayards with this marking that I have seen did indeed have "L.S.K.16." followed by another number (as does yours) on the front grip strap. The only difference from Stewart's description is that his text reads "L.Sk.16.", rather than "L.S.K.16.", but I believe this is likely merely a typo. His speculation that these guns are from the Greek trials is based on the serial number range of specimens examined with the "L.S.K." marking, rather than any suggestion as to what "L.S.K." actually stands for.
I find Still's explanation for these markings a bit weak. Yes, if the "L.S.K." marking were on an arm issued by the Imperial German Army, then the expert on Imperial German markings would be the best interpreter of those markings. But just because an abbreviated marking could make sense in German doesn't mean that it is indeed German. Many other countries and organizations used a similar format in marking weapons.
Isn't it highly unlikely that a batch of foreign pistols, whether captured or purchased, would have gone through the armorer to be unit stamped, and then issued to a unit of troops, without ever being stamped with a single Imperial German Army acceptance marking? I think it makes more sense to infer that these guns, and the "L.S.K.16." marking are not German at all.