I use the Robline and it will dye up nice. Only 2 of the 3 strands take dye so you end up with a white speckle in the solid color.
RIT clothes dye is what I use.
I believe the Robline that Rich is referring to is their "Racing Sheet", which is a nylon/dyneema blend. As you have noted, Sta-Set is a polyester line, so RIT may or may not work. I think your plan of testing a small piece sounds like a good one- I'd also run the piece through the washing machine to make sure its colorfast before it bleeds color all over you/your boat!
Let us know how it goes, I'm curious to see if this works.