XAML Islands for WinUI 3 is not ready yet : 1-under-consideration
(some XAML controls can be embedded like I explained in this thread, but it is just a workaround and not really usable)
You should do the reverse, Win32 APIs in a WinUI 3 app
(as unpackaged WinUI 3 apps are standard Win32 apps, you can do anything a C++/Win32 app can do)
From what you wrote in other threads, you can put it in background (just no window), add a Tray icon with Shell_NotifyIcon, etc...