"It's Tan, so you have to account for that."
Dye is weird. It reacts very differently with other dyes and colors and fabrics. The wrong dye and the wrong fabric and you have a mess. Those links I posted give good info on that. Be careful with the bleach. It may not touch the color and give you something worse. It could give you a green or orange that you could never dye purple. There are plenty of places to cut a swatch from on the coat and TEST first. Under the collar, in a pocket, etc. Any spray paint or paint you use is gonna be stiff. That much "Fabric Paint" is gonna cost an arm and a leg. It's not a white fabric. You'll either have to paint it light and then purple, or soak it down with the purple. If you do that, do it way ahead of time, because there's a chance it'll never cure on it. Either way it's gonna be stiff. Textile Airbrush Paints may not even stick to it if it's not cotton, they are way cheaper and less stiff, but again it's not white.
With no idea about what the fabric is I really can't help you. We gotta know what this "felt-like" material is. Don't know if it was Khaki or a Caramel, or Butterscotch color. If it's wool or whatever. What do the care instructions say if there's a tag in it? Who's the Maker? Post a picture. It's blind conjecture on what would be best.
Laffo