You just have to open your source, that part which depends on Qt. It's not a real problem. But get a commercial license anyway, the cost is small compared to the other costs of developing your program, and you want to be friends with them.
(There's someone on HN who lives on a single-line modification of an open source program. Trust me, source availability of the source code of your client app won't really make a difference.)
Now I want to know more about this :-)
Using a shared object/DLL is the traditional way of doing so, but you can also accomplish this by providing the object files for your application to allow users to link their own substitutions statically.
The FSF explicitly permits this as documented here:
https://www.gnu.org/licenses/gpl-faq.en.html#LGPLStaticVsDyn...