If the system wide "Dark Mode" is enabled on Windows, Kleopatra's tray icon is not shown. It is shown after switching back to "Light Color Mode". Tested on Windows 10 Build 22H2.
KIconTheme sets the fallback theme name to breeze, but those icons cannot be found because we only load the icon theme RCC for breeze-dark. I think we need to load both RCC files in dark mode. No, that doesn't work.
I think we need to add all breeze icons that have no equivalent in breeze-dark to breeze-dark before creating the breeze-icons-dark.rcc.
One more data point: breeze-icons installs a copy of all breeze icons that do not exist in breeze-dark in the breeze-dark icons folder. So, with icon files on disk breeze-dark has all icons that breeze has even without using breeze as fallback icon theme. Looks like an oversight that the breeze-dark RCC generated by breeze-icons doesn't include missing breeze icons.
And another note: In KF6 icon inverting happens automatically in ksvg or so, so that we don't need to ship breeze-dark anymore. And there will be a BreezeIcons library including the icons that can be used instead of the RCC file. This means we just need a quick fix for VSD and not a general solution for upstream.