Except that the world wouldn't learn Booster is a genuine hero because there would be no reason or excuse for Barbara to tell anyone. There would be no harm in Babs knowing after the fact, especially if it's okay that Bruce or Dick know. And given that both Bruce and Dick were made aware and that was okay with Booster, it would cause Babs to lose even more agency and further use her as a useless prop instead of a fully realized character in her own right.