This. If it's that important to Judge let him pay for it. I'd sell to whoever was offering the most money, maybe offer Judge last right of refusal.
I've seen estimates that put the value of it around $100,000 to $150,000. $100,000 is less than what Judge was paid to play that one game.
I wonder if these fans that catch special balls are highly pressured into giving it back. And are told something like "We'll make it worth you while". Then after giving the ball up, they're given a shirtsey or whatever.