Yes, the C-stock would have been a replacement on the early Remington M1903.

I purchased an 03-A3 a few years back with the C-stock. It had the K-stamp inside the cutoff recess (Keystone maker). It was also marked with a Circle-P firing proof, and different letters and marks fore and aft of the trigger guard.

I posted on CMPicon forums, wondering why a replacement stock would have the firing proof, and John Beardicon identified the stock markings as being consistent with an overhaul at the Remington Armory, versus the other well known rebuild facilities. Perhaps your numbers and letters were applied by different inspectors than for mine.