Chad,
Not so sure that the food coloring would prove anything. I'm no chemist, but since food coloring is made to adhere color to other things(e.g. cookies), it seems like that happening in your experiment just would show that food coloring does indeed dye things as intended. The food coloring could produce the residue. It wouldn't prove that without it, a residue is left on LPs.
I'd also be hesitant to try this because I wouldn't want to get food coloring inside the tube. Cheers,
Spencer