As I understand it, the two clipboards I am using in Linux Mint are the X11 and GTK clipboards.
No, not exactly.
The buffer used by middle-click paste is called the selection – not the clipboard. There is only one "clipboard", and that's the one you access via Ctrl+C/Ctrl+V.
However, both are a generic X11 feature – both use the same "selection" mechanism under the hood. When you select some text it goes into the
PRIMARY X11 selection, and hitting Ctrl+C puts it in the
CLIPBOARD selection. That is, 'PRIMARY' is called "the selection" and 'CLIPBOARD' is called "the clipboard". Neither of them is more GTK-specific than the other.
(In fact, when support for the latter was added to GTK in 1997, it was already known as "the X clipboard" at that point, and had already existed for several years before that – e.g. it can be found in Emacs source code as early as 1993.)
But at the same time, both are used explicitly by GTK – neither of them is actually "automatic at X11 level". When you select text, it is GTK – not some X11 magic – that puts it in the 'PRIMARY' selection, and when you middle-click it is also GTK that pastes it from 'PRIMARY', in the exact same way that it handles Ctrl+V to paste from 'CLIPBOARD'. (There is even a GTK setting to disable middle-click paste.) The same applies to Qt, Motif, or any other toolkit.
Is there a way to get Notepad++ to use these two clipboards independently?
No. As mentioned, it is the specific UI toolkit, not X11 as a whole, that must implement the middle-click copy/paste. But because Notepad++ is still a Windows app, it doesn't have any access to X11 selections – the only thing it knows is "the clipboard" that Windows apps use.
Its text widgets haven't been programmed to paste anything on middle-click – and couldn't have been, because there aren't any Win32 API functions to read the "selection", only the clipboard. That means there's nothing that Wine could emulate, either; it would have to invent entirely new functionality to make that happen.
So at best, if Notepad++ is scriptable, you could make it run
xsel -o or
xclip -o as an external program (I think Wine programs can run native Linux programs?), which will output the contents of the 'PRIMARY' selection, and have Notepad++ insert their output into the current document.