You won't get too far when you don't compile your Qt yourself. For a simple Qt GUI app you need at least the Qt5Core and the Qt5Gui dll. They are not so big, However, the available binaries have a dependency to ICU. This lib is a bit large. If you don't need QWebKit you can disable ICU. There is probably some other stuff you can disable when you don't need it, but ICD is by far the biggest dependency.
And don't worry about the LGPL. You don't modify your libs. You just don't deliver the whole package. This is not what the LGPL demands. So it is perfectly fine just to deliver Qt5Core and Qt5Gui. Even when you configured and compiled your Qt5 yourself. You did not change the code of a lib.