Wall of text incoming.

TL;DR on KJTr bet big, bet small, x/r, and x/c are all necessary lines to take with AQ sometimes and I am seeking to explain how all of them can be equally valuable.

First off, for everything I say in this post there will be a exploitative/nemesis way to play better than this if we know leaks in our opponent's game. However I feel like this is a situation where a number of strategies obviously present themselves and I wanted to explain why I would, in a vacuum, want to play all of them sometimes despite having essentially the same hand every time.

Imagine that we open HJ and are called by BTN and flop comes KJTr.
Sample HJ opening range: AA-44,AKo-ATo,KQo-KTo,QJo,AKs-A2s,KQs-KTs,QJs-QTs,JTs-J9s,T9s-T8s,98s-97s,87s,76s,65s,[50]33-22[/50]
Sample BTN flatting range: QQ-44,AKo-AJo,KQo,AQs-A9s,KQs-KTs,QJs-QTs,JTs,T9s,98s

Further, let's imagine we hold AQ, and that BTN also has 16 combinations of AQ. This is a board I would describe as "dominated" by AQ; if great amounts of money get shoveled into the pot we will quickly reach a point where both players have ranges of only AQ and no other hands.

I want to explore four ways to play AQ here and explain why all four of them can be right given our range. Let's start with the most immediately forthcoming one:

1) We can bet AQ for three streets for the maximum amount of value. This will actually require surprisingly small betsizes because villain's range is 13% AQ. I'm not going to go through solving exactly what the betsize this range wants is because other factors like board changes and us wanting to balance it slightly with hands like JJJ and TTT are going to change it beyond what I can work out on my own. If we bet 2/3p on each street villain will have to felt ~21.5% of his range if defending 1-A. As villains get wider in their calling range, or 3bet AQ sometimes, or have exploitable folding tendencies, the betsize for this range will change.

There would be very little wrong with just playing AQ like this all the time, it's certainly the most obvious and sensible way to play it. That said I believe there are quite a few other very sensible ways to play it. The next one is to think about how we can play AQ to maximize the utility of the hands we have to bluff with.

2) We can x/r AQ and balance this x/r'ing range with QQ as a bluff. The point of this play is that x/r'ing against a villain will narrow their range much faster than simply betting will. On this board QQ is, in my opinion, an extremely ideal candidate to x/r, as it loses pretty much nothing if the flop checks through (in fact this is probably more desirable than it going into its x/r lines I would guess). It also has absolutely amazing blockers to villain's calling range as soon as we narrow that range to AQ only, and this is the fastest way to do so; betting would require a larger monetary investment and would lose us money when we had AQ anyway because we want AQ to be getting called by worse. If we x/r a range including AQ, QQ, and a couple of sets, we will likely be maximizing our combos of QQ. As such this isn't so much an argument for playing AQ like this because AQ is a good hand to play this way, but for playing AQ like this because QQ is a good hand to play this way and QQ needs us to do so with AQ as well.

So, what other hands do we have that want to play a different way which we can support with AQ?

3) We can put some combos of AQ in the line we take for value with most of our sets. I would argue that on this board we actually want to bet KKK generally because villain's calling range is not particularly top-pair heavy (lots of QT, QJ, JT, JJ, TT, sure there is some KQ and stuff too but it's less pronounced than on other boards) and because villain's calling range narrows to sets+ quite quickly, and because many hands have 8 outs against KKK. So, here's another place we will definitely want to balance with AQ. Whether we choose to take a smaller 3-street sizing or take the same 2-street sizing and check/decide river is up to us and I think both options have merit.

4) We can put some combos of AQ in our x/c'ing line to defend our showdown-bound hands. Again, this is playing AQ a different way because some other part of our range needs us to. If we x/c KQ and QJ on this board but never x/c a set or AQ we will run into a lot of trouble giving up tons of large value bets unless we fold more than 1-A. Having AQ in this range forces villain to value bet smaller with all of his value hands (like with our AQ we needed to bet ~2/3p because of how quickly we narrowed villain's range to AQ only) which means our KQ and QJ can get to showdown more often and lose less in the process if villain has a better hand.

All of these lines want to have some sets in them as well so that we can still play on paired boards, where the texture of the board instantly changes drastically. I think this is a very interesting board and hope that my analysis on it has been useful to you, I'll leave the actual decision of whether, how, and when to apply the different lines up to you :).

A parting point: if we take these lines with the right frequencies and right balance of partnered hands against a nemesis villain every one of them will have identical EV. Bet bet bet gets max value playing for AQ, but bet small bet small bet small gets extra value because we get raised more often, x/r gets more value because we capture more stabs, and x/c gets more value because villain valuebets worse and bluffs. I have absolutely no idea what the right frequencies are though. It's also technically possible that one of the strategies might be dominated, like theoretically x/r might always be worse than x/c or bet here, but that is not something I know how to prove one way or the other and intuitively I don't think it is the case.