I honetsly know much less about the LSAT than the GMAT, but from what I've gathered, it's easier to do better on the GMAT. In terms of overlap, there's definitely a decent amount, but I would be very careful about burnout.
I don't think that your status as an applicant in one program affects the other. Each school will evaluate you based on your credentials and decide whether or not you're a good fit and can succeed in that program.
Keep in mind the work experience expectations of law schools vs. business schools. Most law school students go right into law school after college, while most business school students have at least several years of work experience before entering school. Just be mindful of this since it may be hard for you to craft a "one size fits all" application strategy between b-school and law school.
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