"To be honest, I'm not sure anybody actually needs a reason to play basketball."
"Do you really need a reason to play basket?"
Both look "okay".
To me as a native speaker, however, the nuance of the sentence-ender 「かね」 is considerably more important for a deeper comprehension of this sentence ...
Perhaps it would help to rearrange it into a more standard word order?
This と is just the standard quotative particle associated with 約束する.
In the lyrics she just said 約束する first and followed that with the content of the promise.