I'd focus on a select number of t14 schools and then schools from the next rung down (i.e., the Fordhams of NY and the GW's of DC) in the cities where you most likely envision yourself practicing. You should get a hefty aid ride from somewhere. Though it's not an absolute science... I got 75k from Cornell and only 30k from BU.
Without knowing the relative chache of Fulbright TAships vs. Fulbright scholarships, I'd say You have a good shot at aid at any t14 school except HLS, YLS, Columbia and NYU. You might get into one or two of these but I wouldn't expect much aid to be forthcoming.
A good resource for aid info is www.lawschoolnumbers.com
. If you search by schools you can get a rough estimate on what sort of numbers will net what type of aid.
UC-B, Cornell and Duke might be your best options for more substantial aid within the t14.