Potassium Permanganate is the way to color the stag. After coloring it, it will look awful, and you will think you messed up. But, after clean-up , sanding, and buffing, it looks great. Try it on some scrap antler first.
You can find it at places that sell water softening/treatment chemicals and pool supplies. It is used to remove iron from well water.
Leather dye works, but the color never looks quite right to me.