A mat might be kind of bulky to pack. This is more what I'm thinking of;

http://www.amazon.com/Slip-X-Solutio...+bath+stickers

I'm not promoting this product or site, just showing you what I was suggesting. There are plenty that aren't cutsy too. A pack or two would fit nicely in a suitcase. They're cheap enough that if they're too awkward to take home, you could toss them.

The shower head slid down to the lowest point works fine with a tot. Some kids don't like the water coming down from so far up but it depends on the kid.

In case I wasn't clear, a parent has to be there for a child that age in all cases. They don't necessarily have to be taking a shower but that works too. I also recommend the "hand-off" if the other parent wants to continue the shower. The other parent can dry and diaper them more easily.