@AndréHenriques Dear Professor Henriques, thank you very much for your wonderful suggestion! I have seen something along those lines considered in the works of Ip, Ponsot, Teschner. However, I can't change representation space and I can't impose the additional constraint you have mentioned. The problem I have described in the post has certain origins, which are not allowing me to do so.

@NicolasHemelsoet True! My naive guess is that it should be something known as dense modules. See, for example, Volodymyr Mazorchuk "Lectures on $\frak{sl}_2(\mathbb{C})$-modules", Section 3.3. But even if it is true, I don't know how to rigorously obtain this result.

@NicolasHemelsoet Dear Nicolas Hemelsoet, thank you very much for your question! I have also asked this question myself when approaching the problem, hence, decided to add the answer to it to my original post.