I am going to say you are wasting time and money to even try dying a synthetic saddle. Those kinds of fiber, smooth vinyl material, just don’t take color like natural things, cotton, linen, leather. You would be better off having color stick if you painted it. NOT RECOMENDING painting it, but at least the color would stick for a short time, unlike dye. Would probably end up VERY ugly, instead of just an unattractive shade of brown.
Part of the appeal of synthetics is easy care, you could probably clean your saddle with a power washer (often), with no ill-effects to the material or color. Not needing the babying of leather, continuous care to keep it soft and nice to the touch.
I would just ignore the color, leave it alone, and use the heck out of it.