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I received a grade I deem unsatisfactory for the quality of work I completed, due to lateness penalties. The course instructor was partly responsible; he has never responded to several of my assignment inquiries. His non-response added several extra hours to my work, prompting lateness on said assignments - enough to lower the letter grade.

Furthermore, the professor has given zero feedback on any of my essay or research project submissions in this course of 'Engineering Ethics', only grades - and has remained 2-3 weeks avg. behind on grading.

I filed a complaint along a grade appeal to the Department Chair, describing said concerns, and proposing a resolution in form of raising the letter-grade (from which I am 0.2% away) - after failing to find a resolution with the professor. The chair denied the appeal. I forwarded the matter to the Dean of Students, who referred to the Vice Provost.

The response from all of them, in a nutshell, was "Policy was followed." None of them made any response to the complaint - which was detailed, and comprised majority of my letters. This was hardly a surprise*.

This said, there's zero intent on behalf of the administration to bulge on this matter. The only college official I'm yet to contact is the Provost - who I doubt will respond any different unless I shift gears.

That policy was followed in grading is true, but overlooks the fact that my grade suffered directly from the professor's negligence - unsure how to press on this point further (or pursue others).
Short of a lawsuit, what approach is advisable to increases the odds of winning the grade raise?

'* - the institution is ridden with incompetence: professors often lack basic understanding of course material, lab manuals are filled with faulty and ill-grammared instructions, outdated website, and a 'sheep'-like student culture that takes any sh*t from professors (hence the lack of accountability).

Worth noting, I called out the chair directly in my letter to the Dean, accusing "complicity with academic negligence" - and stated, unless [college] is willing to risk being exposed as a scam institution, it should focus less on doing cover-ups and instead address the problem."

The first step is to file a formal complaint against the instructor. When your university-internal avenues have been exhausted, you would then need to hire an attorney to sue the university. There is virtually no chance that you will succeed with a lawsuit. If the university had failed to follow its own procedures, or had egregiously violated your rights, you might win such a suit, but the chances of that having happened are virtually zero.

The essential problem is that there is no specific contractual right for you to receive an particular outcome in a class (based on my knowledge of rules & regs in various universities). For example, you might reasonably want to have written comments on assignments given within a week: but there is no rule guaranteeing that you will ever get written comments. (You need to look at the university rules to see what is an actual rule, as opposed to a "goal" for an academic unit). Since there is no such rule, the chairman has limited authority to penalize the instructor (he might for example decide to not assign that class to Prof. X in the future). The chair would not have the authority to raise your grade under the circumstances. As far as the courts are concerned, the university's judgment and resolution of the matter are final, unless they simply failed to respect your rights (exhibited racial prejudice, refused to follow their own procedures).

In certain contexts, professorial negligence could be legally actionable, but that would only be, for example, if an EE professor physically harmed a student by negligently confusing milliamps and mega-amps. His (non)action is not negligent in the legal sense.