Problem Statement

I'm using Sway as my "window manager" in my daily workflow. I happen to like Wayland and am used to tiling window managers, so it's a good fit. I also think the KDE project makes excellent software, which I like using in my day-to-day computing. There, however, is not a well-documented way to make qt or KDE applications use their KDE style outside of KDE itself.

What doesn't work

• qt5ct (or qt6ct). Don't get me wrong, they both do what they advertise, but they don't really let you use your beautiful breeze(-dark) theme in non-KDE environments.
• LXQT. My favorite desktop environment has its own styling/themeing support which is not the same as KDE's. It allows you to create quite beautiful qt setups, but if I want dolphin or okular to follow breeze(-dark), I'm out of luck. This is what I used for a long time to work around my lack of KDE style.
• XDG_DESKTOP_ENVIRONMENT=kde. Ooo, now we're getting close! This technically does what I asked it to do, but it has unfortunate side effects. Various things start to break as xdg-desktop-portals get confused as to which variants they should be using. As I'm using Sway, this needs to be XDG_DESKTOP_ENVIRONMENT=wlr since the compositor and portal base is not kwin but rather wlroots.

What does work

I can not find this referenced anywhere with my meagre ddg-foo, so I'm setting it here for future wanderers to find. After exhausting what I could find online, I went back to the qt5ct setup. I wondered, is it possible to set QT_QPA_PLATFORMTHEME to something else sensible? Yes! It just so happens you can. I discovered, by trial-and-erroring my way though it (breeze, Breeze, Fusion, and plasma all fail), that you can set it to kde, allowing your qt and KDE apps to follow the themeing you set up in ksytemsettings. Lovely good time!

While it wasn't my answer, I couldn't have found it without qt5ct and the reddit community's guides.

Here's hoping someone else can pull out a little less of their hair than did I. I didn't have enough to start with ;-)