Is it possible to restore the plastic on a pickguard without removing a signature on it?
Please do not soak the pickguard in peroxide! That's highly likely to remove or damage the permanent marker signatures since peroxide is a bleaching agent. You might be able to carefully clean around the signatures using a mostly dry small tool with peroxide or other cleaning solution on it, but even then you would run the risk of some of it bleeding over onto the permanent marker and doing harm. I wouldn't recommend even trying it!

Instead, since you're planning on framing it anyway, consider adding another layer of glass between the pickguard and the glass in the frame. Then stain that additional glass layer strategically with a thin wash of paint or ink or dye to visually even out the pickguard that will be beneath it. Colors can be powerful. Try washes in various colors and shades of the same color to see how the pickguard looks through the stained glass. The eye can be fooled nicely with the proper colors.

The advantage to that approach is that you can experiment without any risk to the pickguard, and you can clean off what you did if you don't like it and try something else (glass is very friendly that way). You should be able to achieve the desired result without changing the pickguard in any way.