It's not about whether or not she was ultimately shot and paralysed, it's that that event did not originally involve Booster Gold, and now it does. Therefore, Booster Gold played a (albeit ineffectual) part in a life-altering event in Barbara's life. It involves her in a serious way, and the decent thing to do would be to tell her about it.
I can totally believe Booster not telling her, given his need for secrecy, but Batman is a close friend of Barbara and has enough respect for her that he should feel obligated to tell her.