I don't see why you can't just use a fabric dye. Plenty of options for that.
Here's a type of dye I've seen in other threads that people use on their uniforms:
It's called "Dharma Fiber Reactive Procion Dye". They have it in multiple shades of black.
There are many choices for fabric dyes, though. That was just one. Amazon will have plenty. I think you could even use tie-dye dyes, if they have one in just black.
I've never done this, so it would be interesting to see how it turned out for you.
The main caveat to doing this is that you have to be careful not to let the dye spread across the line of the collar. To ensure that, I think you're going to have to paint it on by hand using a small paintbrush, at least at that line. The rest you can just soak a little (not too much, or it will bleed across the edge).
It's going to take some experimentation. Start on the back inside collar area, and close to the middle of the collar where a mistake won't matter too much. Put a dab on and leave it there for an hour to see how far it spreads. From that, you can get an idea of what it will do. Then do it for real. It might require multiple coats.
Alternatively - and I don't recommend this - see if you can use a Sharpie marker pen. Put a dab on somewhere, let it dry, come back and reapply a couple times, and wash your uniform a few times to see if stays on. I doubt it will produce a very deep black color, and it will probably fade over multiple washings.
Hope that helps.