If you have excess to a airbrush I think spray paining one Liquitex Acrylic Paints is your only bet in this regard.
Black is impossible shade to dye over any synthetic wigs. You end up with a tone of grey always.
Yo could also do it white a fine mist atomizer but it will take time as you want each spray to cur before adding another so one layer every day or 12 hours. (The airbrush make finer mist which is why you can work faster but you still need the paint to cur).
I also recommend doing a first layer in white to help block out the black a bit more.
You also want to comb the hair out after each cur do that the fiber don't get "glue" together.