I would be 4betting here. I don't think you want to go to the flop 4-way with KK. Against 3 other players, you're likely to have around 45%.

I would be 4-betting to try and get the pot heads-up.

At full ring, I have seen players jam or call with such weak ranges in these spots because they don't understand pot odds and "there is just too much in the pot".

Yes, you'll run into AA sometimes and yes, they'll outdraw you but I think this is a clear 4-bet and KK should never really form part of our calling range here.

i don't play a lot of FR, but it seems tough for MP2 to 3b 8% and only call down 1% - is that vs shoves or all 4bets? also is this in this particular spot with two callers or in general? The fact there is a squeeze situation should widen your range and I'd think some villains would adjust for this, no?
If he's calling committal 4bets this tight in general, can't you print money by 4b 3x every time, or 4b shove any Ax you open given blockers etc?
In this particular spot, it's also about the BB etc. I'd assume a 66/21 type is also calling off with QQ, AK+, maybe some JJ or TT if he feels you're squeezing too liberally.

For the 4b size, I'd aim to find a size that keeps it HU yet keeps villain's ranges somewhat wide. Maybe $7? (worried about a "normal $6" bet leading to a calling train).

Although I just play 6-max, where players stack off more liberally, I think you have a clear 4-bet all in shove. Consider the alternatives:

KK is much to strong to fold. So folding is not an option.

Calling, in my opinion, is not a good option. If you call, MP+1 will probably also call. So you were up against three opponents. Additionally, postflop MP+1 and MP+2 would have position on you. That is not a nice spot to be in with KK. If you do not flop trips or better, you have to play the guessing game postflop.

Since there is already so much money in the middle and the stack to pot ratio is already very low (round about 5$ in the middle before your action and round about 15$ effective behind), a small 4-bet does not make much sense in my opinion.

So the best option should be a 4-bet all in shove. Your categorization of the other players favours this play even more. MP+2 has an 8% 3-bet stat. So probably he is capable of making some 3-bet bluffs. As a squeeze play, his 3-bet percentage might even be higher than 8%. So your KK should do pretty fine against his 3-betting range. If he has AA, that is bad luck. But most probably, he will not have AA and fold in most cases or call with a hand that you dominate. Additionally, as you said, BB seems to be a recreational player. When you shove, it seems not too far fetched that he will find a ridiculous call with, let us say, 88, 77, ATo or K9s. (MP+1 as a recreational player could also find an “aggressive call” with a very questionable hand.) So, with that much money already in the middle, and your info on the other players, 4-bet shove all in should be by far the highest EV play that will make you the most money in the long run.

generally 4betting is the accepted play after seeing a coldcall unless you have information that the 3better only has AA/KK and only AA/KK here (in which case maybe you should fold?)

Familiar with MP2? If the 23/9 guy (MP1) is truly this weak/passive, his (MP2's) squeeze can be a bit lighter here. The historically bad BB who flats this spot does not concern me. If he flats AA here sometimes, so be it. I expect to see more TT/JJ/QQ and AK from him than AA. I expect him to call a lot of jams. Ripping here will probably be most effective and also appear to be a weaker range than a range that 4 bets to $6 or $7 here.