Okay, this isn't what you want to hear but....
Dying is iffy, dying anything black is iffy,dying synthetic fabric is iffy, dying a garment made from mixed kinds of fabric is iffy.
If you don't care if it all goes wrong then give it a shot, otherwise don't bother. The cleanup from dying is a pain, especially if you use the washing machine. The dyed garment may not be fast and dye may come off on your skin or other clothing. Sod's law says that this rubbed off dye will be just about impossible to remove!