Dear Rabbi Kornfeld,
1. The discrepancy between the various Gemarot in the Song may stem from different Minhagim IN THE Beis Hamikdash! Could it not be that different Batei Av and Mishmarot had different minhagim?
If so then the different customs prevalent today may each have its source in the Beis Hamikdash!
2. Why have a special song? "The reason is the same reason that our Gemara (54b) gives to explain why the Shir of Musaf of Rosh Chodesh that falls on Shabbos is recited before the Shir of Shabbos. It is to publicize that Beis Din is certainly correct in their determination of the date of Rosh Chodesh. Likewise, on Rosh Chodesh the Shir of the Tamid is changed to show the people that the Beis Din is correct in their establishment of the day as Rosh Chodesh. On Rosh ha'Shanah, too, the Rabanan enacted that a special Shir be recited at the time of the Tamid, in order to publicize that Beis Din is correct in their establishment of the new month."
Question: How does the singing show that the Beis-Din was correct? All it shows is that this day the singers were instructed to sing this song, perhaps by the very Beis-Din which was, ideed, mistaken?!
Suggestion: in quite a few places (Eruvin 32a, Mishna Ketuvot Ch1 M5 bavli 12b, Yoma 66a Rashi DH Ela Sheasu Hakohanim Keva) there is explicit mention of a Beit Din Shel Kohanim, and in others the existance is implied (Rosh HaShana Ch1 Mishna 7: VeKiblu Hakohanim Oto v'et B'no, Zevahim 100a). Thus: The proof that the Beis Din was correct in determining the day was: That the (Beis-Din of the) Kohanim agreed, and THEY were the ones to instruct the singers.!
Mehera Yibane Hamikdash
Yeshayahu Hakohen Hollander
1. The Beraisos and the Mishnayos imply that there was an established practice in the Mikdash, and that it was not dependant on the custom of the particular Beis Av or Mishmar.
2. You are correct that the enactment to sing the Rosh Chodesh song at a particular time does not show that the Beis Din is correct. Rather, the point of Beis Din's enactment is to show that Beis Din is confident and not doubtful about their declaration of the new month. This is also implied by the words of Rashi here (DH Leida).
Your suggestion certainly seems possible as well, although it is a Chidush to say that it was the Haskamah of the Beis Din of Kohanim which gave credit to the declaration of the Beis Din ha'Gadol.
All the best,
M. Kornfeld