How can I effectively challenge a final grade?
I would go through the process set out by your university. Talk to the professor first. You don't have to phrase it as an attack on the TA. Just tell the professor that you feel the grade was too harsh, and you would like him or her to review it. If you feel that the professor doesn't fairly address your concerns, then take it to the department head, and perhaps ultimately to an independent review committee. Typically, in order to get a grade changed by a committee, you will have to show that there was an error in grading your paper or that your paper was not graded consistently with the other papers. If the TA graded everyone's papers harshly, then there's not much of a basis for a complaint. As was pointed out in the comments, if you go to the department head before you talk to the professor, the professor will likely be annoyed at being blindsided by an issue they were unaware of. Also faculty are given broad leeway in grading their classes as long as it is done consistently. Unless there is an egregious mistake and the faculty is uncooperative, it is not the department head's place to interfere, and they most likely will just leave it to the professor.