Villain is an aggressive reg. He, and I, and one other reg at the table have been 3betting and 4betting each other pretty aggressively from late position opens.
Relevant in this hand is that villain can be quite sticky and does not like to fold to Cbets.

I'm in the BB with KK

Preflop
folds to villain on the BU, BU opens to $0.62, SB folds, Hero raises to $2.11, BU calls $1.49

Flop ($4.08) 794
Hero bets $2.24, BU calls $2.24

Turn ($8.32) 5
Hero bets $5.70, BU calls $5.70

River ($19.20) 3
Hero has $14.95 remaining

The case for checking: The only potential value targets I can think of are A9s, TT, and maybe JJ. Villain has 77, 99, and maybe 44, 55, 66, and 79s (I'm not sure how much weight to give to these last 4 combos). They should have some missed draws to bluff with and may even bet worse for value, so to be clear I am planning on check/calling.

The case for betting: I have an overpair and missing value here seems criminal. Many of villains nut combos I listed above may have folded preflop or on the flop.

seems like an easy shove, especially oop

I doubt villian has 55, 66, and there are only 2 79s combos which he shouldnt really have anyway. I would bet $6-7 looking for a call from 9x, TT, JJ and fold to a shove.

With that sizing my call on the shove would be $8-9 to win a pot of $40-41. I think folding in that spot would be a mistake. Maybe I need to size the flop bet bigger so that I can shove river for a smaller fraction of the pot myself?

You should shove if you bet. As a rule of thumb, never bet less than shove with <psb, check will almost always be better especially ip.

Even with the price i doubt we are ever good if he shoves over us when we have that little behind. Maybe shoving is better but we are running out of worse hands that can call.

Thats how I felt at the time and why I opted to check. I also do generally agree with what @Ojune said about only shoving our whole betting range with <psb behind.

Yeah maybe. Betting is for sure better than checking. Hard for V to have bluffs and nothing worse bets.