My buddy summered at Weil DC and loved it. I know it is different than Weil NY, but it seems like culture is pretty consistent across the firms.
Realistically, you shouldn't place much emphasis on ability to make partner. It will be extremely difficult to make partner at any of your choices (I know Latham, who obviously isnt on your list, only has like a 2% rate). Most people from the firms in that level, as I am sure you know, lateral down to be partner.
Congrats on the great options though!