Doubt it will be cheaper than tile, but I've seen it done.

Check with local granite countertop fabricators. You may be able to get a great deal on "scrap" pieces. Rather than putting something on top of the sill (carpenters call it the "stool") you can skip the wood and use any (correctly sized) stone or a solid surface product like Corian. I would avoid quartz, it doesn't "touch-up" easily.

An acrylic solid surface will stand up to the nails, but I'd suggest lighter colors and a matte finish.

Choose your trim color for the rest of the window millwork based on the solid surface. If your Corian fabricator can make you a bunch of standard blanks that could be trimmed a bit by the contractor it would probably be the most cost effective option, but your contractor would also have to know how to handle the material.

You can replace specific damaged pine or MDF pieces 200x over for what you will end up paying a fabricator to create custom Corian moldings for your house. Or pay a dog trainer to help you create a less indulged canine good citizen with manners.