Ask the paint pros at your home improvement store (bring a photo). They can show you the best paints and application methods for this.
Right, there is a paint that is made to stick to tile but I haven't tried it myself. My daughter did in here bathroom and it looked great. I think she said they had to clean the tile well to remove soap scum build up, let it dry completely and they they painted. Paint did take some time to dry and cure before the could introduce moisture to the room so I hope you have another bathroom to use in the meantime.
We painted the tile in our rental property. It's all about the primer. We used this https://www.sherwin-williams.com/homeowners/products/extreme-bond-primer
Here is how they turned out. http://alifeunfolding.com/blog-1/2018/8/25/updating-the-kitchen-in-our-rental-property?rq=kitchen
This one also has an update one year later which I always think is nice...I like knowing how things hold up! https://www.hometalk.com/7059591/painted-bathroom-tile-one-year-later