Hello guys, Im working on that part of my range let's say we are in BB and we 3bet a BTN open, flop is 422 rainbow.

Now to simplify the spot a bit, we have in our range all broadways with AA-TT and he has 99-55 and broadways without AK.

What would be our best cbetting (and barreling?) range in a vacuum ?
To try to answer this question here are my first thougts :
Cbetting 100% even if our range is really stronger than his doesnt looks like the best because our range isn't as strong that we could afford that. So what part of our range should we check ?

I often see goods players checking here AK/AQ type hands and try to get
them to showdown. So lets start with that. If we only checks thoses
hands here they cannot stand too much heat, our opp could bet with all his pair as nuts aswell as with some weaker hands as bluff. So if we check/fold all our checking range at some point, we better only take
the lowest part of our range on the flop and X/F it right here (JT/QJ type hands) and bet with the rest.

However if we want to protect an AK/AQ checking range we should check our nuts (overpair) a % of the time. The first problem with that is it will weaken our cbetting range allowing us to bet less bluff hands. The second problem is that we almost never get 3 street of value with them. He may double barrel a % of his range but not 3 barrels it especially if he sees that we check AA sometimes.

So I would like to get your insight on thoses points : Why should we decide to check AK/AQ type hand compare to the nut low of our range that we could easily X/F and do we need/how we do balance that ? How can I construct the best cbetting range/checking range in that spot ?

Thanks!