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Hero: A K
stack 3900

Blinds 100/200, 2 mins until 200/400 in a turbo 20min/levels rebuys still active for 2 mins

10 players
Fold to fishy MP+1 (equal stack), limps
Fold to Hero in the CO, Raise to 800
Fold to MP+1 who calls

Stacks:
Hero: 3100
MP+1: 3100
Pot: 1900
Flop comes
K Q 8

MP+1 checks

Hero shoves, MP+1 calls with K J

Stove puts AK ahead by 53/47, I feel like I could have bet 1/2 pot and laid the hand down on the turn if a diamond came, but the crux of my question is really the most +EV play to make. I don't think I'm thrilled to ever be flipping this close and putting my life on the line, but in a turbo I really don't have the time to mess around. I don't see where I can really maximize my EV in this situation, and this situation comes up A LOT so I would really appreciate your advice.

Definitely don't like the plan of bet/folding with TPTK here especially with the rebuy period still going. Given that it's 20bb (and 16on the flop) looking for excuses to fold is probably indicative of some serious leaks. If the structure is really fast then the value of staying in drops significantly.

I don't play many re-buy's so I am not used to the dynamics of these.
With these stack sizes I would either push allin preflop or do what you did and min raise (perhaps a bit higher to make the flop about a pot sized shove) to induce and just shove the flop regardless. If just called preflop when hoping for the induce I would expect quite a wide range and keep playing accordingly so with the momentum a shove would seem fine.

It would be a terrible mistake to raise fold the flop here! If you bet 1/2 pot and get shoved on the pot size is 1900 + 1900 = 3800, and you only have to call 1350 more for a final pot of (3100 + 3100 + 300) = 6500

The equity you need to break even is (bet to call)/(final pot size) = 1350/6500 =21%
Unless you had seen the villain's cards you should call with almost anything at this equity.

Once you raised preflop you were pretty committed to going all the way especially in the wilder rebuy period.

I suspect you should try to plan out the future of the hand more as far as stack sizes and betting goes.

I think the most +ev play would be either to:1. push preflop or 2.Make your raise sizing a little smaller.if you can make it lets say 500 preflop the pot will be 1300(6.5 bb) rather than 1900(9bb).This gives you room to c-bet fold and still have a playable stack .e.g. in this situation if the flop comes j 4 7 giving him top pair ,you can still c-bet 400 into 1300 and fold ....u still have 3000 in your stack whereas with the raise to 800 preflop your c-bet might be more like 750....here you would only have 2350 left in your stack having invested 1550(just under 40% of your original stack ). You would almost be pot commited to calling villains push with ace high ...

We have 19BBs, the blinds are about to go up and it is a turbo, so I'm not looking to keep the pot small pre so I can get away post.

I'd just jam pre. If it gets through we add ~12% to our stack and we will be looked up by worse some % of the time.

Edit: I tend to consider that if I shove with a hand as strong as AK it also balances my range since I will be pushing weaker hands in spots like this too. Is this good logic or am I silly to consider balance at low stakes? If I don't need to be balanced, what is optimal here with 19BBs and AK?

We want a premium hand this close to the end of a rebuy period. I see premium and think what is the best way of getting as many chips in as possible.If i think a shove will get called then shove it is We can double even triple if were lucky.There will be many times when a lesser hand gets it in with us, worse aces can turn up with alarming frequency. A double up followed by an addon gives us a great boost leading into the next periodwhere the blinds are getting steeper and antes are in or soon.

IMO the biggest mistake here is to raise to 4x the bb because you should get the same results with a minraise usually if you want to steal.
So: if you steal w 4x bb then you are paying too high of a price for the chips you get and you get into a lot of what might be difficult situations post flop that you don't need to take in order to succeed. If you shove pre you get the most folds (which is necessary for AK).
Maybe it is not such a big mistake to raise 4x when a possible caller is calling a raise and folding most of the time even with these stack sizes, not sure.