Carl Darling Buck (October 2, 1866 – February 8, 1955) was an American philologist. (wikipedia)
To a certain degree, yes you can bargain. The real key is how much value the institution places on the individual applicant.
It's far easier to deny an appeal if it's just a letter, or an email. It's always a roll of the dice, but in person, the probability of success is much greater.
You may have two students with the same EFC but the student with the higher grades gets more aid.
You have a consumer decision to make, but you don't have a frame of reference to work with.