Will mixing red and blue food couloring work or is there a better method?
Yes. Buy purple food colouring. any baking supply store will have it, and there are plenty of suppliers online. Powdered colour is much more intense than liquid.
You might be able to get that color with a blueberry juice or maybe beets, but purple food coloring would be easiest.
I think that for casual home use, mixing your own blue and red is a perfectly fine idea. I've done it for making playdough and it is just fine. You could experiment with a few drops in water to determine the blue-to-red ratio, but unless you need a really specific color (Minnesota Vikings Purple or somesuch), just using equal amounts is fine.