mAdmAn

Because you want to be check/raising merged on a low/medium flop with intermediate strenght hands, when the opponent is cbetting a flop his range doesn't hit as well as yours, to protect a pair for example from eventual high cards to come so you won't let villain realize his equity so easily.
But here, you should not raise merged something like A8, because you are both holding two pairs or sets, he has some TPTK you can't have, so this middle pair is not weak enough to be turn into a bluff and not strong enough to raise for value. You want to check/raise polar with big hands or big bluffs you carefully chose, and just call intermediate hands.

As mentioned before, 3bet shove on the flop is the best option to me, even though it's a really huge bet. But as played, I think you have the odds to call on the turn. Unless you know for sure his raising range on the flop is only flushdraws, he might have some straightdraws as well. Here is a reasonable raising range we could attribute him:
AQo, AQs, KJs, JT of spade and diamond, T9-87 diamond.
Facing this sizing, the call is +EV. And the range I gave him here really tight. He could actually be balancing semi bluffs with value as good as with this range, but we could also find more bluffs. KJo with the king of diamond for example, 3 more combos. The 2 others JTs combos. It would gives you 42% equity, more than enough to make this call profitable.
Anyway, this is a tricky spot because it depends a lot on SB's range here, wether he is able to call 1 gappers or not to 3bet bvb such as T8 or 97. Again, easiest way to get out of this is to shove on the flop.

I agree on the sizing with both previous comments, I'm not sure you want to be check raising with this hand on this flop though. You are clearly raising polarized here, because it would not make much sens to be raising merged on a K high flop against HJ (MP), so you do want to go big on the raise flop and on the bet turn, but you don't have that many hands to do so.
Obviously you want to be raising with K8, K7, 78, 88 and 77, which makes 13 combos of pure value, and you have to be balancing with some bluffs. But raising A6 of spades here might lead you to be overbluffing this spot.
The first natural bluff would be T9, I don't know if you are defending T9o against an unknown profile in HJ, so let's say you only defend T9s. 4 combos. Then you have clubs flushdraws. AT to A2, A8 and A7 excluded makes 7 combos. You might also want to raise QJ of clubs, JT or J9 which gives you a lot of equity to go on. You would already have more bluffs than values. If you also defend 96s or 56s against HJ, you would surely like to raise these clubs combos as well.
So now, if you are adding A6 of spades, A9, QJ, J9, JT, I think it might be too much.
My point is that you should probably work on your flop raising ranges because it will keep you away from these complicated spots, where you do pick up some equity with A6 when the turn is a 5, a 9, or a spade, but end up with too much bluffs afterwards in my opinion.

Anyway, you talk about 24 combos that you're happy to 3 barrel but you
never mentioning checking any of them on flop or turn. Do you consider
the need to protect your checking range? If not, you should look into
it and make sure to check Jx a certain % of times on flop/turn (small
%, but very important to have)

Well, I think I do need to get value as soon as possible with hands that at least have trips here (KJ, QJ, JT and J9). The only hands I could check back within these 24 combos are AJs (1 combo) and AJo (4 combos), because I would be blocking everything I could possibly get called by. This is why I was thinking that a big bet would make more sense, betting the nutted hands and some bluffs such as KQ or T9 with backdoor flushdraws, and check back AK, AQ, AT and so on, so then my checking range would be protected. On the 238 combos my HJ opening range contains, there are only 68 combos of pocket pairs (KK and QQ included) and 12 combos of suited connectors that did not hit the flop (87, 98 and T9), the rest of it has at least paired an ace or get a straightdraw. So is it really necessary to check back some of the Jx when I have 80 combos of Ax (AJs and off excluded) I could check back?

I disagree with the part where you say you should never bet AK here.
Multiway we should be betting quite often here. BB should have a very
wide range and missed often. SB can be more condensed but will pay
with lot of worse hands for sure.

Yeah, I meant I should never bet AK with this sizing because getting polarized with this hand here doesn't really make sens, I'm not bluffing but I'm not really sure that I'm going for a lot of value either, because TT or 99 could call a 1/3pot bet, when they will surely fold against a 3/4pot bet.

Is your range way stronger than theirs? BTN open is wide so we have
lots of hands. SB call is condensed around broadways and pockets so
will connect well. BB is wide but will have lots of Jx and Ax. Sure we
have AA and AK. But they have plenty of Ax and Jx.

I'm sorry about that it was not very clear but I was opening from HJ, not from BTN so my range is really tight from there. Except some suited connectors (76s, 87s, basically) and pocket pairs that missed this board really hard, most of my range is doing great on this flop.

Smaller bets is just a thing you do in multiway pots. You can get away
with a lot of bluffs this way for a cheap price and there's not much
the other players can do about it.

From the BTN I completely agree with you, but is this thinking the same for HJ? Because SB or BB can't really be floating even against a 1/3bet pot from the HJ on this texture I think.
Anyway, thank you for your answers, hope to read more from you.