I'm pretty sure there is leeway in the appeals process for handling this case if it's absolutely obvious the Jury didn't follow the law.

this isn't the jury "not following the law". it's called "jury nullification" and it has been a feature of common law for hundreds of years.

You can appeal against a conviction, but unless new evidence comes to light, a non-guilty verdict closes the case. At least that's my understanding.

The non-guilty verdict is final, even in the face of new evidence: “[N]or shall any person be subject for the same offence to be twice put in jeopardy of life or limb” (Fifth Amendment to the U.S. Constitution). Sometimes the government gets around this by inventing a new “offence” for the same crime, often at a different political level; for example, you may be acquitted of murder at the state level, but be charged with a “hate crime” at the federal level for the same act.