I dont think Columbia (or any US Univ wil have a point of view that they dont want to take in students who would like to work in USA. They do have a point not to take in too many US students who did their JD rom lower tier schools.. do not want to encourage people to upgrade a lower rated (or lower gpa) JD by doing an LLM from cLS/HLS? Other higher rated schools.
Secondly, scholarship vs reputation of school - at higher rated schools you have lesser chances of a scholarship or waiver (and significantly lower availability too), but a lower rated school may be more pliant.. but then what happens when you get into job market afterwards? So, it is really a trade-off decision. For me, an admission in CLS/HLS/top5~10 without aid will have preference over a full fee waiver at a lower tier school..
And yur green card makes it easy for you to get a job later, and also to get finance from a bank..

Thank you for the advices! Any other kind of Scholarships available from Private Institutions or Non Profit institutions?