Unsatisfied with my previous attempts, I did a little more research into the subject. I ran across a youtube video that did a pretty good job of digging deeper into the subject, and some of the challenges that it presented.
The biggest issue at hand is that there is a fundamental split between how indexed palettes are traditionally handled, and how modern GPUs approach rendering. Essentially, it's a matter of integers versus floating point values. GPUs, and their various shader languages, are all about those floating points. While classic scan-line rendering, and the palettes that helped to fuel it, were primarily based on integers and integer calculations/logic.
Writing modern shaders requires that we focus our data and calculations on floats. But traditional palettes are almost entirely centered around integers. The challenge faced is creating an effective and performance-efficient translation layer between the two.
After much research and a little trial-and-error, it's pretty clear to me that selecting colors individually on the Material level is probably NOT going to happen. Any such assignment will likely have to happen through a GDScript. I will go greater in-depth later.
In Unity, I was able to utilize array structures from within the shader language to implement some of this. In Godot that will not be possible, at least not with the current state of the shader language. I thought of trying to jury-rig some matrix definitions for a similar effect, but that seems like it will be more trouble than it would be worth. Fortunately, a different solution presented itself, which I believe should offer a reasonable alternative.