How can I make suction cups stay securely to a tiled shower wall?
You could definitely just silicone the suction cups to the tile.  This is kind of permanent in the sense that you will not be rearranging this, but not so permanent that you are hurting the tile.  To take it out a putty knife would do the trick.  Might have to use a solvent to get rid of the remnants but not a big job at all.