Master's degree at target school (from non-target undergrad) for consulting?

I will graduate at the end of this summer from a non-target school with a degree in economics, and my objective is to get into MBB or a Tier 2 consulting firm within a year or so. I was originally a pre-med biology major before I recently switched to econ, so unfortunately all my experience is clinical (research, hospital volunteering, etc.). The closest thing I have to business-related experience is a econ paper that I'm working with a professor to get published.

At this point, I have two options - I can attend a one-year masters program in Health Policy at the London School of Economics, or I can spend the next year doing internships or a full-time job if I can get one. Which route would make me a more attractive candidate for top consulting firms?

Some things to consider;
- I have pretty good grades (3.9+ GPA) and good test scores (2300+ SAT)
- I am open to working anywhere, both within the US and abroad (I plan to come back after a few years for business school)
- Although I will be paying the LSE tuition out of pocket, money isn't really a factor in my decision
- Because of the way that the LSE program is structured, I can probably fit in part-time internships during the year, and the program curriculum includes an internship for next summer.