I go to an unknown (and therefore completely lacking in prestige) law school in the US. If I achieve stellar grades in my JD and pass the bar, would an LLM from a prestigious school help mitigate my JD being from a low-ranked law school?

if you are talking about a tax LL.M., and you get outstanding grades during your first LL.M. semester to complement your great J.D. grades, the LL.M. will give you a small boost. It may not be fair or logical, but even graduating top 5% at an unknown school is much, much worse than graduating slightly below median at a top 20 school. The LL.M. won't sanitize your unknown J.D., but if you do great in tax, make connections, and do great work at your first job you can build a great career notwithstanding your unknown school.