When I was a teenager we used to dye my friends cats and dogs. We used to dye our own hair weird colors(pink, blue, green) and we found out the hair dye we used was safe for animals too(called the company) as it was made from vegetable dye.
It was pretty funny dying the cats but it wasn't what I would consider stress free. You have to put the dye on, leave it for 30mins while you make sure the cat(or animal) doens't try to lick it off. Then you have to wash them. Which with a cat was fairly stressful, as the cats did not want to be bathed, lol. It was hilarious seeing cats and the dog dyed a bunch of different colors and patterns but I would guess this way would be WAY too stressful on a rabbit.
If you really want a dyed rabbit, I would do the food coloring/spray bottle method. It won't give you a solid colored rabbit but it should still tint them the color.