What I meant about the LRAP deal was that some schools do not cover you if your job title is not "attorney". For example, a policy or legislative analyst job that requires a JD, but doesn't involve actually practicing law may not be covered by some LRAP progams. Foreign Service, etc. may also not be covered.
If you are serious about your career path, keep in mind career services may not be able to help you much, even if your school has an office or counselors dedicated to public interest placement. The reason for this is that very few people from even the top law schools land the type of jobs you're talking about, so the career counselors don't really have any useful advice to offer. Academic advisors are pretty much worthless, IME. You'd be much better off tracking down alumni that have the jobs you're interested in and speaking to them about how they got their jobs. Also, your school might have a student org up your alley, with the occasional guest speaker that might be able to give you some good info.
I doubt earning a dual degree is going to help you much. Dual degree programs, with rare exception, are just money makers for law schools.