In theory, I do not see why this would not be possible. There are plenty of aquarium-safe plastics.
In practice I would be very hesitant (just like you). Products for human sanitary use tend to have chemicals hidden in them that are completely harmless to us but can do a number on your aquarium.
Phtalate and BPA free is a good start, but I'm also worried about things like fungicides - a lot of bathroom products contain small amounts of fungicides (because fungus would of course love to grow on your bath mat). They're usually harmless, but when submersed under water for long periods of time they can be quite harmful to aquatic life.
I assume the bath mat would come with a full list of "ingredients". Just sit down and google them all - it's often very easy to find chemicals and materials and the effect they have on aquatic life.
Rinsing and soaking is always a good idea. A lot of fungicides and the like are (while very lethal) also very bio-degradable. That means if you put the mat in fresh water for a day a few times, most of the harmful stuff will be gone.
It's definitely worth a try, but something I would be very careful with as well.
