It is possible to successfully negotiate a higher scholarship, I did. Please keep in mind that I have very limited experience with this, as I've only done it once. Anyway, here are some pointers:
1) Like any bargain, you need to have something that the other party wants. You mentioned that you will likely receive a large scholarship from this school anyway, so I assume your LSAT/GPA numbers are well above their medians. This is to your advantage. They want you because thay believe you will pass the bar and be a successful alum. Someone who is offered a, say, only a 25% scholarship probably has far less bargaining power. The school simply doesn't want them as much.
2) A comparable or higher scholarship offer from another school will give you leverage. I had a 75% offer from one school and something like 50% or 60% from my first choice (it's been four years, I honestly don't remember). After writing a letter explaining the situation my school of choice matched the 75%.
3) Be absolutely 100% honest, do not exagerate or embellish anything. If applicable, send a copy of the offer from another school(s) with your letter to show that you're for real.
4) Be positive. I wrote about all of the things that I truly loved about my school and why I wanted to attend.
5) Be humble. Remember, you're asking them for money, not the other way around. Don't try to convince them that you're the biggest badass in the world and that they're fools if they don't give you more money.
I hope some of this helps. Law school is absurdly expensive, and in my personal opinion many law students would do themselves a service by focusing more on scholarships and less on rankings. They usually have no clue how utterly crippling a $2500 per month loan payment can be. If you're talking about a truly elite school, yeah it might be worth it. But for the vast majority of mid to low ranked law schools, no way.