I solved my problem and learned some things on the way:
1. The interpreter is given as a std::unique_ptr<tflite::Interpreter>. The special thing about unique pointers is that the objects they point to don't get deleted once the constructor has finished, but only when the pointer gets deleted.
2. You not only have to save the interpreter pointer in your class but also the resolver and the model.
So if you store those three items in your class as variables you can instanciate them in the constructor and use them in your class functions.
Hope this helps someone