An invoice provides supporting evidence for that value, but even invoices can be challenged.
A invoice by a private seller has no more information value than the customs form on the outside. Except for the term "invoice".
In my case it was a common lens. In case the officer assummed a fraud, 10sec of googling would have showed him that the price on the customs form was identical to the average US Ebay price of that lens. End of story. (By the way, it was a 150Euro item)
As a good citizen I am willing to pay VAT on both the good and shipping cost and was prepared to do so, as of course I calculated that in. However I do not understand why one is gaggled by authorities to the verge of loosing a lot of money.
In this case at the last day the offcer accepted a bank document of mine stating me to have transferred a sum the same as on the customs form to Pay-pal.
In another case, a large parcel full of old, but mint (60 years old publisher warehouse stock) technical books from the USA, german customs lost or held the parcel 3 months! at Bremerhaven customs without informing me or the local customs offce. So that I and the sender considerd the parcel lost and I already ordered other copies of those books. When the parcel arrived it had been opened but not taped again so all books were sticking out.... I am not willing to take that authorities may handle goods that bad and still are able to put losses at the risk of the citizen.