When Better Placement is turned ON and CarryOn is installed, taking entity causing to drop it right next tick.
It feels like Better Placement overrides some CarryOn code, since without this option, picked up entities not dropped untill click again.

There is a video demonstration. I'm pressing and holding Shift + Right Click to take a cow. Normally it would only take a cow, but with option enabled it will drop it next tick, and then take again.

Does this occur if Force New Location=false for the Better Placement tweak?

When Force New Location=false entity will be picked up and placed indefinitely:

I tested CarryOn by itself without UT, and it seems the 2nd clip is the default behavior. It looks like 1.12 CarryOn, unlike its modern versions, doesn't have the proper checks to prevent the behavior of constant picking up/placing while holding the button. The only reason it seems like it works in the 1st clip is because of unintended interaction with Force New Location=true. You can easily check this is the case by continuing holding the button while looking at a new block.
So there is a bug when Force New Location=true because it doesn't match default CarryOn behavior, but the entity being instantly dropped is a problem with CarryOn itself.

Can reproduce, but I'm having the same issue with Blocks, but not entities, which is super annoying bc Carry On is 95% about picking up blocks.

Can reproduce, but I'm having the same issue with Blocks, but not entities, which is super annoying bc Carry On is 95% about picking up blocks.

Actually true. With Better Placement picking up blocks is even worse - when entities placed back next tick, blocks placed back next frame, which means you need to release mouse button in 10ms or block will be placed down.

The old Better Placement mod does it much better