I’ve done edge lit pieces a few different ways and painted acrylic worked fine. I prefer matte spray paint because it’s more forgiving and shows less flaws and spray paint is so easy to apply. And depending on how you intend to light it, you may or may not have to do a graduated engraving for even lighting. I’ve never done it and I think they still look pretty good. 
I didn’t keep a pic of one I fully painted and then engraved, but just to kinda give you an idea, here’s one in the reverse with the engraved part painted black.
And here’s painted acrylic with light shinning from behind. (The teeth had a thin coat of glow in the dark paint.)
These below are wood or black acrylic layered over clear acrylic. I prefer this over paint because I think it looks a little cleaner, but obviously it takes a lot more material. (The last dog one is a different version than the one I posted above.)
