1) A Jew should have a name found in Nach (which is kinda strange, cause as far as I know Chaim isn’t a name in Nach either)
There is the objection you yourself mentioned. There is also plenty of other names that are widely used with no objections that aren’t names in the Torah either. So why “pick on” Shira?
2) Shira is a davar she’ain bo mamash and/or a shtus, and Jewish names should have more significance.
Again, you could make the same argument for any number of other names. Chaim, Shalom, etc.
3) It comes from non-Jewish or secular Zionist influence/origins.
“Shira” means song. Songs are pretty much universal. Jews in the Torah sang songs.
The Wolf