My first thought was the velcro but then I dismissed it because I think it will be hard to clean and will attract gunk. My next thought was some sort of suction cup holder.
We have one in our sink but it does get in the way some times.
If you had a wall handy, you could attach something similar there but I know for me my wall space is pretty scarce and you're on an island which puts you farther from the wall.
I like the idea of the under counter soap dispenser if you have room for that but measure first. The one I have in my house sticks WAY far down under the counter. I don't think my under sink space in my trailer would fit that.