MSM or MBA w/ the focus on non-profit-please advise

Well sir, first of all, Masters in management is usually for people without basically ANY work experience (for the most part). As for which schools to apply for, it's hard to say without a GMAT score (and some more info about you, about what you have done, who you worked for, your achievements etc.)
Round 3 for american schools can be tough, although there are several European schools which have early rounds around this time/ Does that appeal to you?

Also, lemme ask you straight out: What's the rush? As you know R3 gives you a disadvantage. Now, that doesn't mean you can't get in, or shouldn't apply. Not at all. But it's good if you have a reason.