[100NL FR] 3 way pot with KK

I have been 18/14/2 at this table, overall stats are ~16/13/2.2

MP1 is a huge fish, 62/23/1.9
Button seems decent, 14/10/inf over 60 hands

Whats my action/plan for the rest of the hand here

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HAND #1
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poker stars, $0.50/$1 NL Hold'em Cash Game, 9 Players
Hand History Converter by Stoxpoker

Hero (UTG+1): $117.20 (117.2 bb)
MP1: $100.25 (100.3 bb)
MP2: $107.45 (107.5 bb)
MP3: $58.50 (58.5 bb)
CO: $98.50 (98.5 bb)
BTN: $60.70 (60.7 bb)
SB: $112.45 (112.5 bb)
BB: $202.55 (202.6 bb)
UTG: $98.50 (98.5 bb)

Pre-Flop: Hero is UTG+1 with K K
UTG folds, Hero raises to $3.50, MP1 calls $3.50, 3 folds, BTN calls $3.50, 2 folds

Flop: ($12) 9 3 J (3 players)
Hero bets $9, MP1 calls $9, BTN calls $9

Turn: ($39) 2 (3 players)
Hero ?

I think I would try and check this down. Your OOP in a multiway pot with a reverse implied odds hand, and you may not have any redraws at this point. It may add some deception to the strength of your hand, and give you some information about the strength of theirs when it's their turn to act. When the pots get big and it is multiway people will usually play pretty straightforward, especially when the next bet will put them over the commitment threshold. If the fish checks behind and the button bets, I would probably just fold, because he is pretty much committing himself and you could get yourself trapped in the middle. If the fish bets and the button folds, I suppose it comes down to just how bad this player is if your willing to get it in. When it goes check check and we get to see a free river, I guess you can either value bet or check/call a safe card given your reads and maybe induce a busted draw or Jx type hand to bet. Pretty interesting hand though and I may be way off on this, thought I'd give it a stab, curious to others thoughts/suggestions as always.

This really a spot where I just don't know what the best line would be. I think i would tend to fire a second barrel here, but that may be spew.

I'd be really interested to read what Chuck, Zach or Jagsti do in such a situation.

A decent button calling this flop is only doing it with a few things, basically draws. If he's got a set/TP he pretty much has to raise here with all the draws. The flush draws got there so I'm thinking we're beating 8T and TQ. I would definitely check turn, and I think I'd call an MP1 bet, just because it's so often going to be over-valuing TP. If button bets a decent size though I think we can even fold this turn. If both players check turn I think we can go for some thin value on the river (assuming blank), folding to a raise.