I set up an Ubuntu 24.04.1 VM to play around with this, and in its default configuration, even the Flatpak is using the GNOME color picker.
There is probably a much better way to do this, but I did figure out a “hacky” way to make it work, though - if you use the environment variable XDG_CURRENT_DESKTOP="kde" with the Flatpak, then it kind of tricks it into thinking it’s running on Plasma, and uses the KDE color picker. The easiest way on GNOME to do that is using the Flatpak application Flatseal:
(For what it’s worth, Plasma comes with a built-in method of changing Flatpak permissions, environment variables and things like that 
)
With that set, and with restarting KolourPaint if it was already open, you should get the KDE color picker:
It’s a little laggy in places, which may be due to the mix-and-match of components going on or because of the VM I was using, but it seemed to be generally functional.
Somebody else may very well have a much better way to make that work, but maybe that will help a bit for now?