I'm just supporting the initial inference and having some validity. It is by proof false, but I wouldn't berate you for thinking it to begin with.
My guess is that part of the problem here is that you're looking at a colored LED which is emitting a very specific frequency (or specific frequencies) that make our eyes perceive them as intense colors like blue or red or whatever. But they're not a "full-spectrum" blue which would contain other frequencies as well. I think this is also the reason I have such a difficult time photographing my green translucent buttons with the green LEDs behind them. The color (to the camera) doesn't look anything like what my eyes perceive.
It's the same reason fluorescent lights look "white" to our eyes but will look green or magenta on film. Fluorescents are made to be perceived as white by human's eyes/brains, but they're not really emitting full spectrum white light. I don't know anything about LED manufacturing, but I'd bet money that they achieve their colors in the same way.
I think if you wanted to get a purple result, that white light (not a "white" LED) going through both red and blue filters would provide what you're looking for.
Or even one of those blue "party light bulbs". It would be dim, because the loss of light through the blue bulb and then through the red ball would be significant, but I'd bet that putting a blue light bulb behind that trackball would give you a purply glow, albeit dim.