Personally I would not stick the UTH to the tub because it makes cleaning the tub a pain. I have always taped mine (paper backed duct tape) to the surface the tub will be sitting on. That way when I take the tub to the washroom for cleaning, I dont have to deal with the UTH. This method also allows you to swap the tub without having to reattach the UTH.

As far as bungie chords go, I would instead use black office clips around the perimeter of the tub / lid.