Here's what I would suggest as an alternate to a shower-mat:
There's all sorts of similar products... some come in strips, some come in dots, etc. The up-side is that they are permanent and don't get nasty. The downside is that they are permanent, and may not be as comfortable to stand on.
As far as fixing the existing damage... Well, you might be able to find a spray-enamel paint that matches the existing tub surface. I've never done it though, so I'm not sure.