1) There is another issue here also. This is what the Pnei Yehoshua asks in Rosh Hashanah 31a (DH b'Gemara Sha'ani): where in the Torah is there any hint to which songs they should say every day from among the Tehilim that King David said? The Pnei Yehoshua writes that it seems clear that the chapters of Shir that were sung in the Beis ha'Mikdash were a Takanah of the Anshei Keneses ha'Gedolah, or possibly an enactment of the Prophets. Therefore, the Kilkul mentioned here cannot be a Kilkul mid'Oraisa, because even if one would say that singing a Shir at the evening Tamid invalidates the Korban ("ha'Shir Me'akev ha'Korban"), whatever chapter of Tehilim that the Leviyim sang over the Korban would be valid mid'Oraisa, because mid'Oraisa it does not matter which Perek they sang. The Pnei Yehoshua writes that before the Takanah was made concerning which chapters should be said, they were accustomed to singing another song, such as Shiras ha'Yam (Az Yashir) or Shiras ha'Be'er (from Parshas Chukas) or any song from the Torah that they wished.

2) Therefore, the Pnei Yehoshua writes that the problem here was not that the Shir affected the Korban. The problem was that they had not said the correct chapter according to the Takanas Chachamim or the Yesod Nevi'im. See also Chikrei Lev, Orach Chaim #50 (to Hilchos Tefilas Arvis #237, page 301, DH Ach), who writes that the Shir that they said in the evening was only mid'Rabanan, while the Shir of the morning was d'Oraisa. According to this, one must say that the Kilkul in our Gemara is that they did not say the correct chapter according to the Takanas Anshei Keneses ha'Gedolah or the Yesod Nevi'im.

Kol Tuv,

Dovid Bloom

1) The Pnei Yehoshua, that I cited above, writes that it may be that the Psalms sung on each day in the Beit Hamikdash were Yesod Neviim; founded by the Prophets. This can be compared, for instance, to taking the Arava on Hoshanah Rabah, on which the Gemara Sukah cites an opinion that this is Yesod Neviim. We can understand that the Yomtov of Hoshanah Rabah, whilst it might not be different than the other days of Chol Hamoed according to strictly Torah law, is nevertheless an extremely important Yomtov of which the most crucial part is the Mitzvah of Arava. Similarly, the Rabbis wished to protect the Yesod Neviim of singing the correct chapter in Tehilim on Yomtov, so it was worth making Rosh Hashanah into 2 days, in order to ensure that Judaism continued according to the way of the Prophets, who were the leaders of Klal Yisrael at the time.

2) According to the possibility mentioned by the Pnei Yehoshua that the correct chapter of Tehilim was an institution of the Men of the Kneset Hagedolah, we can compare this to the Shemoneh Esreh that we daven 3 times a day. The words of the Shemoneh Esreh are not deoraisa, but Judaism without saying Shemoneh Esreh is just not Judaism. Or take Chanukah or Puirm, for instance. Chanukah and Purim are not deoraisa, but they became much better known to most Jews than many Mitzvos deoraisa. The Mitzvot instituted by the leaders of the Jewish people in every generation are the very backbone of our religion, and for this reason it was so important to protect the Shir Shel Yom in the Beis Hamikdash.

KOL TUV

Dovid Bloom