Interesting dilemma. Might be more helpful to show some concepts & color combos.
In any case, if the product is red and you make the box blue, then it will probably help it to make the red product stand out against the blue.
A) Maybe add a red detail to the outside of the box (maybe the pull strap) to tie it to the red product.
B) Make the inside of the box red as well. Maybe lined with a different material