The simple answer is that the judges were wrong, and that the rules that allow you to do it are the same rules that allow you to pay a kicker cost when casting from your hand normally. There is no other rule to cite, because the reason those rules apply to this situation is that no rule exists that would prevent them from applying.
That is unlikely to be persuasive to your judges, however, and would not correct their misunderstanding that led to this ruling.
I think your judges missed or misunderstood these two crucial points:
- An alternative cost and an additional cost are different things with different rules.
- The clause "without paying its mana cost" is very common in effects that let you cast something, but is not universal or implied.
I think your judges read "You may cast the copy" and automatically appended "without paying its mana cost" in their heads out of familiarity with the many, many cards that say that, without realizing that this particular effect does not have that clause.
Casting something without paying its mana cost counts as an alternative cost. You can only use at most one alternative cost at a time, and when an effect lets you cast something and specifies an alternative cost, you are required to use that specific alternative cost. Casting something without paying its mana cost therefore prevents using any other alternative cost for casting it.
Kicker is an additional cost, not an alternative cost, so you could use it even if you were casting the copy without paying its mana cost. I think your judges also forgot about the distinction between alternative and additional costs, however, which combined with the above mistake would logically result in their conclusion.
I recommend pointing out these two points of potential misunderstanding to your judges as an explanation of how they might have based their ruling mistakenly on rules that do not apply, and showing them the first ruling on Wizard's Spellbook as evidence that your interpretation is correct.