Offer it to one of the two long-time Shark captains, Thornton or Marleau, in UFA negotiations.
Both those guys are way too long in the tooth to captain this team. I wouldn't touch either of them as FA options unless they're willing to come in on affordable and short term deals which I doubt either will be interested in.
Brown's a better option for captain since he's been a part of teams that don't choke when the chips are down. But he's not a fit for what McPhee says he wants and his contract is not good as he's collecting $5.8M through 2021-2022 season. And the Kings do not have the draft or prospect collateral to get that done.
I'd take Neal and put the C on him, as he's done a fine job in Nashville wearing the A. If he is willing to entertain a non-longterm deal, I would also try to sign him. Barring that, then you have the option to flip him to a contender before the trade deadline. But again I'd try to sign him, I think he'd do a fine job as our captain.