Unless you are taking the quilt out in harsh, unforgiving arctic conditions, a waterproof shell is overkill and unnecessary. Regular three season backpacking in most of the continental US, a breathable shell is fine. Many quilts use a Durable Water Resistant shell which is more breathable (making it easier to avoid buildup of moisture in the insulation). Unless your dog brings a bathtub full of water in with him, you're probably not going to wet out the quilt. I learned after one wash of my down quilt that it is going to take a lot more water to wet out down than you think - it took me a while to get all my down wet in a half full bathtub.
If the shell gets dirty, wipe it off with a damp towel. If the quilt starts to lose loft, here
is how to wash your down quilt without damaging it.
Synthetics don't last as long as properly-cared-for down, but if you are washing it often, and don't care that you will be buying new synthetic quilts every so often, that might be easier.
Also be aware that sharp items pierce nylon, no matter what the fill, and quilts often have lighter weight nylon than manufactured sleeping bags, so won't be as resistant to dog's nails.