In the mishna why does it say that another person stirs the blood if the korban pesach was slaughtered before the tamid shel bayn ha'arbayim? Since the same person who slaughtered the korban pesach is not necessarily the cohen who will slaughter the tamid or sprinkle the blood of the tamid, why can he not be the one who stirs the blood?
Good question indeed!
It seems that the Mishnah should actually read "Echad" and not "Acher." This is the Girsa of the Mishnah in the Mishnayos, and it is also the Girsa of the Rambam (Hilchos Korban Pesach 1:4 -- note that the Rambam in Perush ha'Mishnayos leaves out the word altogether). If the word is "Echad," then the Mishnah is not stating that someone else must do the stirring, but just that "one must do the stirring."
However, the Mishnah still could have said, "u'Vilvad she'Yehei Memares b'Damo" ("as long as he stirs the blood," as opposed to "as long as one stirs the blood"), leaving out the word "Echad" altogether. By adding the word "Echad," it seems that for some reason we would have thought that the person who did the Shechitah or will do the Zerikah must be the one who does the stirring (perhaps we might have thought that the stirring is part of that Avodah). Therefore, the Alternatively, I may have thought that the one who slaughtered the Korban at the wrong time is "fined" to stir the blood, or that the owner of the Korban is responsible to do the stirring, whether he be Yisrael or Kohan. The Mishnah says "as long as one stirs the blood" to teach that it can be anyone and not just the person who did the Shechitah.