My vote would be to stick with the bright orange rope, but perhaps wrap it in a black canvas or duck cloth "sheath" for travelling, for exactly the TSA-neutral reasons mentioned. As for dyeing the belt ends, if they're cotton, it can be done, but dyeing cotton to dark colors can be difficult, as the dye uptake doesn't tend to be even (at least with RIT dye, and I've not yet used the fiber-reactive ones) nor particularly saturated.
Or here's a really zany idea: get a wide cloth belt made of black canvas, which then gets customized by attaching a "pocket" on the back side for most of its length. This would hold the rope. Said pocket would be accessible by a zipper along its length. So the outside looks completely standard, but you can quickly take off the belt, unzip the rope-pouch, extract the rope, then replace the belt.
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Or here's a really zany idea: get a wide cloth belt made of black canvas, which then gets customized by attaching a "pocket" on the back side for most of its length. This would hold the rope. Said pocket would be accessible by a zipper along its length. So the outside looks completely standard, but you can quickly take off the belt, unzip the rope-pouch, extract the rope, then replace the belt.
Does this make me a dork too? :)