General strategy in SB v BB 4bet pot at 200bb

Played this hand today and I just wanted some input from both sides about how it should be played from a theoretical perspective.

For one on the SB what is he supposed to do facing a 4bet with JJ? At 100bb he should shove but at 200bb what? How does his range change? Is it 100% a call with this middling hand? At 100bb is best call hand is AQ. Now that I am writing I think I would rather go polar and eliminate all these middling calling hands I see on GTO wizard charts and substitute them for a full all in range with Ax suited hands. But can you still play polar at 200bb deep like this?

I guess I need someone with some experience to describe their though process here.

As per flop I like range betting every A high board like this IP with with vast majority of hands, I should have made it 15bb or 16bb though.

The turn is where I am a bit confused, I check behind to protect my range sensing I need to have really strong hands here sometimes. And sometimes I should barell my trash like TT, JJ, and suited broadway.

River really made me think, but more from my opponent. I of course have to call being damn near top of range, but I am thinking I am running into QQ a lot here or AQ.

Opponent turned JJ into a bluff. I like the thought process from him as most people are not bluffing here, but is JJ the hand he wants to do it with? I guess he is trying to get me off KQ probably. I'm thinking most of the pool might fold KQ there and if they fold KK too I guess it's a really good play.

But what do you think do I have my analysis wrong?

PokerStars - €0.02 NL FAST (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

Hero (BTN): 226 BB
SB: 241.5 BB (VPIP: 38.33, PFR: 25.00, 3Bet Preflop: 8.70, Hands: 62)
BB: 90.5 BB (VPIP: 0.90, PFR: 0.07, 3Bet Preflop: 0.00, Hands: 2,958)
UTG: 78 BB (VPIP: 30.77, PFR: 7.69, 3Bet Preflop: 0.00, Hands: 15)
MP: 126 BB (VPIP: 28.51, PFR: 21.70, 3Bet Preflop: 1.33, Hands: 243)
CO: 101 BB (VPIP: 19.40, PFR: 16.42, 3Bet Preflop: 7.41, Hands: 68)

SB posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has A♠ K♥

fold, MP raises to 2 BB, fold, Hero raises to 6 BB, SB raises to 16 BB, fold, fold, Hero raises to 33 BB, SB calls 17 BB

Flop: (69 BB, 2 players) 2♦ A♦ 9♥
SB checks, Hero bets 11 BB, SB calls 11 BB

Turn: (91 BB, 2 players) 8♠
SB checks, Hero checks

River: (91 BB, 2 players) Q♥
SB bets 43 BB, Hero calls 43 BB

SB shows J♦ J♠ (One Pair, Jacks)
(Pre 57%, Flop 13%, Turn 5%)
Hero shows A♠ K♥ (One Pair, Aces)
(Pre 43%, Flop 87%, Turn 95%)
Hero wins 167.5 BB