How can I get the eyedropper to take a sample and then choose the nearest color on a limited palette?

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The Dropper tool doesn't work in the same way that you might be familiar with in raster graphics programs. It doesn't really "pick up" a color. I don't quite understand what you mean by the "nearest color", when you can get the exact color. Limited palette....the only way to restrict the Dropper tool to a certain or limited palette, is to be actually using that palette. Click the tiny left-pointing triangle/arrow at far right end of palette, to choose a palette. Or you can create your own unique palette, per image, by using Swatches. The manual will have detailed info about both the Dropper tool and Swatches. Help menu > Inkscape palette.

The Dropper tool can be used to fill an object with the exact color of the pixel over which it passes or hovers. First select the object that you want to fill. Then switch to the Dropper tool, and move the mouse/pointer over the color you want. Then click. Of course, if you truly need to get the color from any specific pixel, you'll have to zoom in to at least 500%. (I usually like 800%, on my screen and system.) Or if you click + drag, you can get all the colors in the red circle....sort of averaged, I guess, although that's probably not the proper terminology. That might be something like what you're asking about with the "nearest color"? Transparency can be a bit tricky to deal with. You can probably get a better understanding of how to configure the 2 buttons on the Dropper control bar, by reading the manual, than by my explanation.

If I haven't answered your question, it might help if you could provide a little more into about what you're doing