First, the one is in the body, and the “paid” comment is a footnote, so there is no continuous flow of words. It is more as an aside.
The court is egoistic
(Now as an aside comment, let me say that: “If, even as the price to be paid for a fifth vote, I ever joined an opinion for the Court that began: ‘The Constitution promises liberty to all within its reach, a liberty that includes certain specific rights that allow persons, within a lawful realm, to define and express their identity,’ I would hide my head in a bag...”)
He’s clearly saying that something was GAINED by being the 5th vote....
I read it as him saying that if there ever was an opinion on a case that he agreed with and the opinion began with a sentence like “The Constitution promises liberty to all ...express their identity,”, even if his was the fifth vote needed to affirm the opinion, he would be so embarrassed by his vote that he would hide his head in a bag.
The price to be paid is the embarrassment that comes from joining an opinion with that particular sentence.