the pink stuff isnt mildew, It is a airborne bacteria called Serratia Marcescens.
It likes damp conditions and is commonly found growing in bathrooms (especially on tile grout), where it manifests as a pink discoloration.
I would put in a bigger more powerful ceiling fan or leave your shower door open to help it dry faster. it also feeds off phosphorus containing materials such as soap and shampoo residue.
it says to clean with a bleach mixture but in your case with having travertine it's not good.
Serratia Marcescens Bacteria
others might have an idea.