It is not a problem to wear a rosary in the bathroom as long as you are reverent and keep it clean.
However, the Rosary is not intended to be “worn for protection”. Jesus and Mary will protect you whether you have a Rosary on or not. The Rosary is not a lucky protective charm and, unlike a medal or a crucifix necklace, it is not intended to be worn. I wear mine occasionally, but only because I am planning to pray it shortly and I need my hands free to drive and want to make sure I don’t lose the rosary out of my pocket. I take it back off when I am home and done praying on it.
Since you pray on your rosary during the day, it’s okay to wear it if your point in doing so is to have it handy for prayers. But if you want a devotional item to wear, I would suggest you get one meant to be worn, such as St. Benedict Medal, Miraculous Medal, or Brown Scapular. Also remember that these are devotionals, symbols of our devotion and to remind us to pray. The medals and such don’t “protect” us or keep us safe. It’s Jesus who does that.