Hi there,
Yes, I would reach out.
A week after the final interview is a perfectly reasonable time to follow up, especially since you have another offer that requires a decision by Friday. Recruiters see this situation all the time, and a professional message will not come across as pushy.
The key is to frame it as providing context, not demanding an answer. Something along the lines of:
"I remain very interested in Bain and wanted to let you know that I have another offer requiring a decision by Friday. If possible, I would appreciate any update you can share on my application status, as Bain remains one of my top choices."
This gives recruiting the information they need. If a decision is close, it may help accelerate the process. If they are still deliberating, they will tell you that as well.