Now before trying the exhausting steps to build Qt statically, I would like to know, whether this approach could solve the problem.
Based on the comments where you mention that becoming a commercial client of Digia is a problem, I would say no.
Static linking cannot happen with proprietary software, and based on your comment, your project is not open sourceable.
That being said, you seem to have your own wrapper anyway, and Qt keeps binary compatibility for a given major version the lifecycle of which is usually several years, so I would personally go for dynamic linking and some kind of binding.