No, insurance usually doesn't cover this kind of incident. Even if it did, it's not worth the points/rate increase; better off taking care of it on your own.
I had a '91 Supra that had to have the front bumper cover replaced on several occasions during the 12 years that I owned it (VERY low clearance on the front end!) But that's it; only the cover needs to be replaced. Having said this, however, the new paint has to have a "flexing" agent added to it and it is NEVER as tough as it was originally! Just make sure that you choose a shop that you know does quality work. This at least limits the problems that you may encounter after the work is done.