Well you can’t stain it, your only hope is finding the magic solution that lifts the clay (and clay is the worst). Bona? Vinegar? Diluted ammonia? Steam mop? Obviously long term a warmer mid tone floor sounds better suited.

Magic Eraser is slowly removing the finish on your floors, and making it stain worse. It's basically sandpaper.

Dirt colored floors are always the best choice. Plus walkoff matts, or a dog wash station on the deck where the dogs feet get hosed and then dried.

I would ditch the magic erasers, yes, they act like sandpaper and are removing the finish on your floors. Find a good cleaner that will work to remove clay. Put down a few runners in the main area. Have a place set up to clean the dogs feet when they come in, are all things, that will help you with your floors.