Making ColorSetter work like a PasteButton

Here is a solution.

I have made 2 modifications on your code:

ButtonPaste[] replaced by Button[] (the Attributes are not the same)
Setting[] (useless) replaced by Paste[]

Full code :

   Grid[{{
    Button[
       Graphics[{Dynamic[col], Rectangle[]}, ImageSize -> 26], 
           Paste[col], 
           ImageSize -> 26, 
           ImageMargins -> 0, 
           FrameMargins -> 0], 
       ColorSlider[
           Dynamic[col],
           ImageSize -> {75, 25}, 
           AppearanceElements -> "Spectrum"
           ]
       }}, 
    Spacings -> 0, 
    Alignment -> Bottom
    ]

Button[] has the attribute HoldRest (ButtonPaste[] has not), so the col (of Paste[col]) is only evaluated when the Button is clicked.

If you wrap your entire widget (as currently shown in your code), in Dynamic, it will work. The reason is that without it, changes is col are not propagated to PasteButton.