BU is a very loose fish 31/20 3bet 13.6 over 52 hands

here is my thought process over the hand:

4bet preflop to isolate for heads-up postflop. shoving allin preflop, to my mind, was not effective as I am gonna be slightly dog against V's Calling range. Calling squeeze was obviously not an option as I was likely to be in multiway postflop (Raiser and Caller were fishes as well).

So here is flop which is quite awkward. The plan might be either hit/fold or shove allin regardless of the board. in my considerations both strategies are valid although shoving looks like more profitable (V's wide calling range which gives high fold equity). But still I am not sure how I should've played the hand right.

I would be happy to hear your thoughts on the hand.

Hero (SB): $30.15 (100.5 bb)
BB: $43.07 (143.6 bb)
UTG+2: $37.08 (123.6 bb)
MP1: $31.09 (103.6 bb)
MP2: $26.69 (89 bb)
MP3: $28.11 (93.7 bb)
CO: $24.53 (81.8 bb)
BTN: $34.87 (116.2 bb)

Preflop: Hero is SB with A K
2 folds, MP2 raises to $0.67, MP3 calls $0.67, CO folds, BTN raises to $2.31, Hero raises to $5, 3 folds, BTN calls $2.69

Flop: ($11.64) 7 4 2 (2 players)
Hero?

52 hands is a meaningless sample size. He may have just had a really great run of cards. Furthermore, I'm not sure I'd call a 31%vpip "very loose." A touch on the loose side, but not a totally unreasonable number.
A solid start

Check
You are out of position, you four bet, basically, as a bluff (a really great bluff, but still just A high.) Continuing here with a bet isn't going to get you anywhere. Consider, what is the worst s/he could have here? Your opponents range is likely full of pocket pairs. Your range has PPs, yes, but also more combos of AK and AQ in it. I believe the range advantage is V's in this spot.

bit bigger pre
bet flop 1/2p w/ 100% of your range to set up turn shoves
shove turns accordingly
on a brick, ship JJ+, AKs/AQs combos with fds, give up w/ AKo unless you spike a pair

thank you for the comment. I agree sample size is minuscule but a couple of hands before V was spotted squeezing w JTs in MP so I assume his 3bet/4betcall range is pretty wide. For this reason AK doesn't look like 100% bluff here as V is expected to call most of the time. So I need to have a feasible plan PF to get as much value as possible. Checking Flop isn't -EV but I think it doesn't get the most value out of the hand, I put in quite a lot pre to give up post. And once I hit a pair on the Flop I assume I get paid off just around 30% of the time whereas V's wide 4bet calling range gives me around 60% FE, to my rough estimation.

thanks for the comment. It looks reasonable and safe, I am not risking a stack at least and shoving Turn w FD is totally +EV given the pot odds. I love it but on second thoughts I think 1/2p considerably decreases FE to my AQ+ range, which is the biggest part, doesn't it?

After I posted my thoughts on the hand, I ran the hand through Snowie. Snowie suggests checking 86% of the time and betting half pot 14% of the time.

I think your estimation of your FE is a bit high. I think Vs range consists almost entirely of over pairs to this board. As much as "yucky" as it feels to give up after putting that much in pre, your V feels the same with a stronger range. So your bet is likely to at least get called. I would expect a raise most of the time in this spot.

I'm no pro, these are just my thoughts.

think you need to bet at least 40% pot to set up turn shoves and it's just a bet size that fits the strength of your range
your range is so narrow and strong you are allowed to bet all combos otf (your value bets increase the ev of your bluffs and vice versa)
this is what solvers like to do, they 3b/4b oop linear ranges (i.e. strongest X% of hands for the positions), favorable and neutral flops for your range are very often cbet 100% or close to that and turns is where you decide what you want to bet for value and add the proper amount of bluffs to make your opp indifferent between calling and folding

Honestly, I haven't used solvers yet and have no clue how they work apart from most of them based on GTO. I do believe they are useful but rather on higher limits. Micro-smalls are pretty much all about exploitative game, imho.

I think the solver gave that suggestion based on V's narrow 4bet calling range. If that's the case then I totally go with that. But I put V widely and give them 22-QQ, JTs, Qts+, KTs+, A2s+, AJo+, KQo. So with that range V is going to continue with 22, 44, 77-QQ, all FD combos, and maybe sometimes with AQo+, 33,55,66 which makes up around 40% of his entire range.

I am not a big expert either, just trying to figure things out.

sounds good to balance a betting size and adjust it to the strength of the range. It also protects my ranges.

I think it's too fancy given the opp and limits though

I guess I don't understand this concept. If something is optimal, by definition it should be used at any point. I think the idea of exploitative play is, at it's core, based on understanding theory. So, you exploit mistakes that Vs make, but in order to recognize an error one must know what the "right" play is. So why not work hard to play a game that is as close to "right" as one can get and allow your opponents to impale themselves on their own mistakes?

I'm interested what action you actually took and to see the turn. I have a guess as to how the action went, but I'll wait.

My understanding of playing micro-small limits is to play for fat value against weak loose players (fish) and do exploitative play against regs. The GTO is always +EV but I think it brings less value at small limits as players are generally bad at those limits and it's much more profitable just to exploit them. Exploiting means to take advantage of their tendencies and turn it into profit (e.g. too often fold to 3bet or cbetting or dubbing Flop and Turn and fold River etc.) Of course careful players may change their play once they felt they were being exploited, then you have to adjust your play accordingly. In a nutshell, it's always about exploiting and adjusting. To counteract being exploited you have to balance your game.

In this case I was playing against the fish and ideally I had just to get fat value out of them which means vbet only with complete hands or combodraws. But w Ak offsuited oop it's quite hard to do it. if it had been 3bet pot I would've played Check/fold on the flop pretty much all the time but in this case it's kind of down the drain play. So on the Flop I didn't like playing Check as I expected V to bet almost 100% and, given their wide range, I decided to get it in hoping to get value from the FE. The opp called showing AdQd, which meant I was slightly favorite but still I was't sure if I played right.
All in all I think in this case it would be better to bet pot, which would've done almost the same job, and Turn C/F unless improved or playing GTO as ionutd mentioned. So I am still not sure.

Matt Hawrelinko uses a great analogy. He says that in Jujitsu (sp?) lots of things work against white belts. Almost anything you do will work. But a lot of the stuff that works against white belts will get you in trouble against better fighters. So why not learn what will work all the way to the top? The correlation to poker is easy to see. A play like that at higher limits will absolutely destroy your bankroll. I don’t care if you want to learn theory. It doesn’t affect me. But saying it’s usesless at the micros is none sense.

I didn't say it's useless. I said it's less profitable as, to my mind, I lose value playing strictly GTO at micros and smalls. The GTO is good against solid players who play balanced game. That's why it's hard to make profit by exploiting them. Of course, if at smalls I get up against regs who play balanced poker, I will have to apply the GTO against them. My goal for small stakes is to beat them as quickly as possible, that's why a win rate is predominant for me. And I am fully aware that at higher stakes I will have to play different poker. But that's just my personal perception.