Sorry it's been a year, but that application seems to do exactly what I want to do (at least from the screenshots), while without one I can't find how. I want the day name besides the clock, not besides the date. Is there a way to do this without that application?
ElevenClock - Download and install on Windows | Microsoft Store remains the most sensible solution. All other methods are clever but have many unintended consequences due to globally modifying the internal Windows short date format. The ElevenClock app lets you write fully custom format strings in your taskbar clock without effecting any global settings.