Hi LiJ I prefer "since" in that sentence simply because the reason being given ("our mistake, not yours") is information that is most likely already known to the listener. I don't think using "because" will actually make it any politer. It seems to me that a "politer" sentence would include some sort of apology rather than a reason.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.