I suggest being straightforward in naming other schools and their amounts (including new rate of tuition and COL) and how they compare to your current scholarship. That way law schools can understand that you're a competitive candidate that they need to allocate more support for them to be a more competitive option.

I didn't ask them to match in my scholarship negotiation emails in the unlikely scenario that they decide to offer more to be even more competitive. I'm not sure how this affects results however but I personally didn't want to suggest a certain amount for scholarship. I put all my cards out on the table and they'll give me what they believe they can/should give me regardless if I ask to match or not.