Thank you for the first hand feedback!
I am notorious for using purses as toolbags, so it's no wonder I see lunchboxes the same way.
I will have to experiment with them to get it right, but right now I'm thinking of double end closed head wrenches for the handles.
The could be part of the set that is being carried, or extras.
Umbrellas as source of cloth is a great idea, I was thinking of canvas or the cloth they make hammocks out of.
I wonder if we
should even call them "handles".
Most pictures show the furoshiki being held by the knotted cloth, indeed the wood "handles" often have magnets in them, so they stick to each other.
They are more like a clasps than handles.
The strap and the hoops are definitely used as handles, but I'm less attracted to those.