It shouldn't be leeching dye.
If it's a genuine, real suede strap I would consider washing it.
If it's genuine suede, without a load of padding stitched inside, then run a sinkful of warm water and add a good capful of some decent fabric detergent to it.
Let the strap soak for an hour, topping up with hot water to keep it warm.
Then put your Marigolds on and give it a good massaging in the warm water. Add more soap as required.
When colour eventually stops coming out of it, rinse it in clean cold water until no colour at all comes out, not even a trace.
Remove from the cold water and squeeze out as much water as you can. Give it a good squeeze in an old towel until you can't get it any dryer, then hang it somewhere to dry naturally. Don't force dry it with a hair dryer or in an airing cupboard or in the sun. Just let it dry naturally.
Visit it frequently to rub the fuzz back up with your hand.
Be advised it may take many days to be completely dry. Possibly longer.
Hope this helps.
Karl