I haven't solved the problem yet, but I know what caused it.
QtCreator is doing what it's supposed to, that is, finding the 5.15.1 libraries correctly.
My problem is that my Cutelyst version was built with Qt 5.12.8. CMake automatically found and used the Qt 5.12.8 libraries I have installed, which allowed the version built with only CMake to run correctly.
So, I built a Cutelyst version manually using the 5.15.1 libraries instead. I'm having trouble getting CMake to detect that version, but that's a separate issue, so I'll close this one.