Just another option here -
The fact that you already have a BCG offer shows that you are generally interested in Bain and Strategy&, or you wouldn't be going to the interviews. Therefore, you may get the chance to flip this and build on it.
Should the conversation come up naturally...maybe in 'Why Bain?'
"I already have a confirmed offer for one of your competitors - however, i have read/spoken/researched a lot about Bain and i actually feel as though its better suited to be - the work you do in X,Y,Z is more aligned to my goals and i think this would be a perfect flit. Hence, I've come along today as i do believe Bain would be my first preference of role but of course, i am here to learn more about what you guys do and if my gut feel is right that Bain would be the best fit should i receive an offer"
This 1- displays subtly you already have an offer and 2- gives your answer even more merit, showing YOU choose THEM, not just the other way like for 99% of canididates who apply for every company and hope to hear from ANYONE. This helps making them think you're generally super excited and keen to work there and 3- like Sidi says, actually gives them a way to sell you the firm.