It's too bad you're not interested in pre-made airbrush colors - because they will give you the best results and performance for your $. The solid colors I use are $4/8oz. and a little color goes a long way.
With close to 20 years of airbrush experience on just about every surface known, I don't DIY colors. Tinker? Yes.
Culinary Airbrush Color Recipes may be hard to come by - it takes SO much trial-and-error to get the mix right that it's just not cost effective. And, there are so many options for pigment - dry, paste, gel, liquid... making pearls/lusters, metallics adds to the complexity. Pigment extenders are mostly water with a bit of liquid sugar (like glucose), maybe a bit of glycerin or alcohol to help speed drying.
That being said, so long as you go work with LIGHT layers, you can build up d.i.y. color to good opacity without running and with good control.
Regardless, cake decorating surfaces take color differently than other airbrush surfaces and almost all WILL absorb color to some extent - so fine detail work is possible, but crisp details WILL soften after application. How fast will depend on the surface.
That's why most use hand-painted lines for details. I prefer layering airbrush with hand painted colored cocoa butter. Others like gel colors mixed with vodka/alcohol.