Fellow med student interested in dual degree MPH, so advice from a peer:
Given your career goals, I would say just go for it. I'm in the middle of conversations with MPH-trained physicians who are involved in public health projects -- they all say there's no best time, just depends on how you weigh pros/cons. IMO, it sounds like an MPH follows the "what you put in is what you get out" principle. If you can see a benefit in your career by having it, then that's that.
Personally, I'm interested in the formal education in public health via an MPH, but I'm interested in going into emergency medicine. Just don't know how to tie in the MPH with my future career just yet.