A little change will make the question more clear. It's the confusion between the phrases along with and together with. With is added there because when you add it, they become idioms.
together with (idiom) - including; in addition to; as well as
along with something (idiom) -in addition to something; in the same way as something
Clearly, if you mean in addition to, both are interchangeable in that context. If I were a supplier, either is okay to understand that you need both - manual and product.
However, over both those sentences, I'd prefer telling this - Please put manual (inside the color packing box) as well.
That's because we are likely to suggest putting manual (as they are likely to forget it!) but it is very unlikely that if don't instruct anything, only manual comes and not the product!