'Let's sing the birthday song' is wrong. Which birthday song? It would be better if we will say; 'Let's sing a birthday song' then it will be followed by one of the birthday songs that you or they know. We could say' 'Let's sing a birthday song, Happy Birthday'. 'Let's sing the song Happy Birthday' is correct. Which song? The 'Happy Birthday' song. Or we can simply say 'Let's sing Happy Birthday'.

A native speaker would probably just say 'let's sing 'happy birthday'.'

Let's sing birthday song together Write like this

I'd say "let's sing the Happy Birthday song" but, it is correct to say "let's sing the song happy birthday". In the last version, it implies: let's sing the song called Happy Birthday". The first one would sound more natural to me, but a native speaker may tell you more.

yes it's !