From my observations, beginning in the mid 138xxx range we see various tiny letters stamped in various locations on the barrel lug, some in the recess like your K --
194858 01 GHE 12-gauge, 1 frame.jpg
and some on the bottom of the lug near the frame size number.
154991 02 J.G. in oval and & pink arrows to tiny letters.jpg
207979 02 pink arrow to tiny W.jpg
152235 02 J.G. in oval and & pink arrow to tiny letters.jpg
While K and W seem to be the most common, letters I've observed include V B K O J C X F W and on some Trojans a number. I see plenty of guns in this serial number range, which extends into Remington guns, that don't exhibit these tiny letters on the lug.
I'm sure they have no relationship to the K and K
f barrel steel markings on the early Titanic/Acme and Vulcan barrels.