Both betting and check/calling are good here. I probably go with a C/C here most of the time though. On these monotone boards i genuinely like my value bet range to be 2 pair plus.

I like to take check call lines with these weaker kicker top pairs on monotone flops. It saves us from getting raised and put in terrible spots and we get bluffed into plenty. If it checks to the river just go for a value bet there.

I also agree that it's totally fine to have this in your check/calling range even against weaker opponent given the board texture.

I think considering the villain, limp on the BTN, not a full stack, i would bet almost 100% of the time against this type of players. Thinking he would probably call with all of his pairs, and draws, i would go bet/bet/bet if i get raised i fold. Flop around 75%? for me no reason to go small, against a probably fish CS

I'd likely just bet flop on the large side. I'd expect a half stacked limper to have a pretty wide range here so I'm happy valuebetting KJ and expect to be called by any hand with a club in it plus tons of worse pairs. It gets iffy if you're raised but I'd probably fold to one given this would be near the bottom of my value range and I wouldn't expect villain to semibluff super aggressively.

What range do you assign the button limper that has him hit 48% of the time?

These limpers have such a wide range it's pretty insane, I don't even want to list it here. I know exactly what hands they limp-call with because every hand is shown in the HH. I'm even discounting their junk in Flopzilla, adding my cards and they still hit this board relatively hard for a limp-call range (44%). Even if I give them a conservative limp-call range and factor in my cards, I get 49%. It's not like they smash the flop, it's mostly an accumulation of stats. Even playing with flopzilla again with this flop - the mono flops are creating percentages in the mid-40's, which is interesting.

I'm cbeting against this limpcaller and I'm cbeting 2/3 or more every time. He's gonna call so wide. And we are losing value by checking, because these players will check behind their draws all day long. No need to give free cards. He can call with weak pairs, gutshots and all kinds of nonsense. We MUST TAKE value from limpcallers.

Limp callers generally don't bluff raise too much even with strong draws so it's an easy value bet for me. As Komodo said villain will call with so many worse hands, Kx, 9x, any club, straight draws and some other pocket pairs, so bet big, 2/3 min but even pot isn't bad. If villain raises just muck it but I really don't like checking the flop, it just gives them a chance to get there with everything we want value from. And if villain just folds then happy days.

Easy cbet for me and I'd make it 3/4. These guys are passive for a reason, they don't bluff raise here often enough to worry about it, but they will call all day to hit something so I'm not going to let them hit it for free.
Punish dem limpers.