The problem is going to be your lemon cake mix is a pale yellow color so if you add red food coloring you will just be making different shades of orange.
I would suggest a plain white cake mix and add 2-3 teaspoons (to taste) lemon extract.
The same for the frosting.
In both instances (providing the mixes are white) you need to start out with 2 or 3 drops of red, stir and then add 1 or 2 drops at a time, stirring after each, until you get the color you like.
Food coloring has come a long way. You may actually be able to find PINK. Just remember the more you use, the more intense/dark your color will be
I have been baking from scratch and box for over 40 years