I've posted a thread very similar to this, but I'm still unsure about how to play these spots.

Example:

We 2-barrel AKo on 89Tcc 2
Whether that's good or not is irrelevant.

River is an A and we decide to make a fairly thin value bet, whether or not that's good.

How big do we bet?

If we bet big, Villian is forced to defend fewer bluff catchers, and he will fold most worse hands, and bluff catch mostly with hands like A9/89.
And it's also not too appealing to get into a big pot with one pair here. Maybe that's a flawed thought thinking.

So betting small for value seems like a solution. How do we balance this? If we were bluffing, we'd want to bet bigger, to make him fold most bluff catchers, so that won't work out.
And if we bet small with thin value hands, and bigger with thick value hands, Villian could in theory check-raise here, because we've capped our range.
Do we bet small with two pair, to balance that out?

To summarize:

1. Should you bet big with a thin value hand, or does it not work because Villian is forced to fold most hands you beat?
2. If you bet small, what about our bluffs not being big enough, and him being able to bluff catch more?
3. Say we do build a small bet sizing range this way, how do we keep that betting range strong enough, so Villian can't attack our capped range?