I don't think SouthernCakes is QUITE correct in saying that most Broadway shows start 10-15 late. Traditionally it's 5 minutes, but I have noticed in recent years that it's been creeping up toward 7-10 minutes. 15 minutes is more extreme for a Broadway show, though not unheard of. BUT it's important to note: once a show gets into their regular rhythm for the run, they will usually begin including regular curtain holds into the estimated run time.
In any case, I think you probably CAN make this work, it's just a question of how much you're willing to put yourself through the stress of it. Even if Doubtfire lets out at 5:50 (which would be very late), you could still make it to the Brooks Atkinson on time, and you could do it by walking at a brisk pace, without even having to run. But there's little margin for error in case something goes wrong. I would recommend wearing a watch, and having a snack on hand! Also consider: will you enjoy the last few minutes of Mrs Doubtfire if your mind is elsewhere? Or the first few minutes of Six, if you're still catching your breath? And if not, is it still worth it?
These are questions you have to ask yourself, but personally, I think it's probably worth it. I know how it is on these theatre-binge trips, and in general I go by the principle of: if you have an opportunity to see more shows, do it. When you look back on this trip, are you more likely to regret not seeing Six? Or are you more likely to regret having to spend 10 minutes rushing from one theatre to another?