A genuine family emergency is a valid reason to ask for another reschedule. I would not force yourself through an interview simply because you are worried about how a second request looks.

Keep the message concise and professional. The same principles in this guide to writing a professional email to a consulting recruiter apply here: be direct, respectful, and clear about the action you are requesting. You do not need to provide highly personal details. Explain that an unexpected family emergency has arisen, apologize for the inconvenience, reaffirm your interest in the interview, and ask whether another date is possible.

There is obviously no guarantee the firm can accommodate repeated changes, particularly when interview schedules are tight. But that is different from assuming the second request automatically destroys your candidacy.

The more important question is whether you actually still want the opportunity. If you are reconsidering consulting because of the travel, hours, or nature of the work, separate that issue from the emergency. Do not withdraw from an opportunity you want just because you feel guilty about rescheduling. Equally, do not keep rescheduling an interview for a career you already know you do not want.

Family emergency > BCG interview. Just tell them. You're gonna perform like trash anyway if your head is somewhere else

Second reschedule isn't ideal but withdrawing when you actually want the job makes even less sense imo

I rescheduled twice for medical stuff and still interviewed. Recruiter was totally normal about it

But you also said you were already questioning consulting. Maybe figure that part out before asking them for date #3 lol