I think (without certainty) the P was not put on a spare but only a stamp from the arsenal.
Many A4s had S stock during the conflict. I had a video of the troops in action on the Rhine and briefly saw a sniper with his S butt.
the first models released had more regularly the stock S and the commercial 330 C weaver scope.
I let the professionals of 1903 answer these questions and benefit from their knowledge