The problem here is not that Verizon can't justify such a fee. Nobody selling anything needs to "justify" their price, not least because it's unclear what that even means. The only meaningful measure of value for anything is what someone else will pay for it. And people are apparently willing to pay $5/month for an unlisted number.
That does not mean that this price is not outrageous. It is. If Verizon had any competitors at all, the price would be much lower.
The problem here is not "price-gouging" - it's monopoly.
Of course you can claim you forgot. But the judge has to believe you or you will get the same punishment as if you refuse to disclose. So the issue becomes how to plausibly forget. Some below suggest complicated two-factor authentication. But it might be enough just to cite the rarity of strong passwords - if they were easy to use/remember, everyone would use them. You do, but, predictably, you forgot yours. It's a stressful time, you know.
Just make sure you never write the password down. Not only because it could be discovered before you destroy it, but because destroying it could be construed as destruction of physical evidence, opening you up to obstruction of justice charges even if you are exonerated.
What ever you want is going to cost a little more than it is worth. -- The Second Law Of Thermodynamics