I MIGHT have a pro-tip for this. Instead of steel wool and nail polish remover, you might be able to use something called Krud-Kutter Graffiti remover. I just used some on my plastic gun butt stock and it removed some caked on paint and didn't affect the plastic at all. It's seriously amazing stuff.
Here is a before and after pic:
and after
It's not very toxic BUT PLEASE UNDERSTAND, I DO NOT KNOW IF IT WILL WORK ON THIS TYPE of plastic! It should be fine as the bottle says it works on plastic among about 30 other things.
If someone has a scrap dash piece they want to try it on, or mail me the test piece, I will try it.
I bought it at Walmart.
Here is the product: