Big Law to Private Equity Realistic Path?

I'm the first in my family to to graduate from a top 25 university and the only person in my family to attend graduate/professional/law school. Needless to say, I had no exposure to ibanking/private equity etc when I was in undergrad and thus, I didn't pursue it at all.

However, I will graduate law school next semester with a minimum of a 3.6GPA  and have accepted a job with a top law firm (i.e. Kirkland & Ellis, Skadden, Davis Polk, Latham & Watkins) in New York City. The plan is to work at the firm for 2-3 years, then exit into MBB. (To this point, during law school I applied for and was accepted into Bain ADvantage and Bridge to BCG, so I hope that this will give me a leg up after school). Then, after two years I want to exit into PE. I'm familiar with finance (undergraduate degree is in Business Admin. from a Semi-Target business school), but, obviously, I would have to work on my modeling. Targeting any and all UMM/MFs in Los Angeles and New York. At the time I leave from my law firm, I will be anywhere from 29-30 years old.

Posting here to get some feedback on my plan/thought process. Is this realistic? Should i plan on getting my MBA?

Thanks in advance!