How do I take effective stack and ICM considerations into this tournament situation with AKo

An interesting hand from an event I played this past Sunday that I would love to get your feedback on.

9 handed

Final Table

10 paid:

9^th: $480

8^th: $560

7^th: $720

6^th: $970

5^th: $1200

4^th: $1800

3^rd: $2600

2^nd: $3500

1^st: $5000

These are approximations, but pretty close to actual payouts…

There is a good chance there will be an ICM chop at some point in final table, as there is no trophy or ring at stake.

Seat 1: 200k

Seat 2: 200k

Seat 3: 250k

Seat 4: 135k

Seat 5: 150k

Seat 6: 220k (Hero)

Seat 7: 150k

Seat 8: 150k

Hand:

Blinds 2000/4000/500

Seat 4 in EP1 opens to 10k

Folds to Hero in MP1 and Hero 3bets with AsKd

I debate how much to raise…looking at EP1’s stack…she has about 135k

A 3bet to 25k would be about 20% of effective stack, a 3bet to 35-40k would put EP1 in a place of folding or shoving…

What is the best strategy here? Would it change with ICM considerations at this final table?

I raise to 35k…how would you size your bet here?

Folds back to EP1 who tanks then calls…

Pot: 80.5k

SPR: 1.2

I now realize I optimally would have bet 40-45k or maybe shoved to get the SPR ~1

Flop:

7d 6h 3s

EP1 tanks then checks

Hero?

Should I shove all flops here?

I place Villains range of

22-QQ, 33% KK, 33% AA, ATs-AKs, AJo-AKo, KQo, KQs, QJs, JTs, 50% QJo, 50% JTo

So my main question is would EP1 have played 89s, 78s, 67s, 56s

I think there is a small chance, but at this point, I would heavily discount, maybe 20% weighted

So basically I am behind the sets (9combos); the 2pairs (we will discount to 1combo) and the combo pair draws (discount to 1combo) and the straight (1 combo)

Also behind 99-QQ, and the 33% weighted KK, AA

I really feel EP1 would shove the 88-JJ here, so let’s discount from 24 to 12 combos

And then there are a ton of overcards about 60 combos

I decide to bet 35k…should I just shove?

In retrospect, I think a shove is the default play, but I believe this player is capable of slow playing a set, and capable of folding to a C-bet of 35k the same amount as if I shove 70k effective all in.

EP1 tanks then calls?

Turn: Kc

Villain insta-shoves.

I heavily weight Villain's range with Kx combos that have hit...

In retrospect, I played this well, and my read was spot on, but should I still shove the flop?

Should I value aggressive play (by shoving flop) over equity play (calling a turn shove) to reduce # of showdowns (and variance)?