You should be able to buy mini-suction cups separate from the mirror. Suction cup tape is elegant stuff and would be perfect for this. It would be a great alternative to possible damaging or staining the tile with an adhesive. It is easy to remove but will stick the mirror up there until you want it down. It is expensive but you will find other uses for a roll.
I don't think you will have problems with the reflective backing blackening on you. In the days when mirrors were silvered or had other metal coatings prone to oxidation? Maybe. Plex is probably going to have reflective mylar or something bonded to it.
As for size. You should be able to cut, or have cut, a piece of reflective plex to whatever size you want. Do you have a real plastics shop near you? They might have a minimal cutting charge. They can also polish the edge for you if that is an issue. Plastic edges are not as bad as glass but if you catch a finger tip on one in the shower, you may feel it unless polished. A plastic shop may also have a plastic more scratch resistant to recommend to you.