Harish-Chandra bimodules of finite $K$-type in Deligne categories

We continue the study of Harish-Chandra bimodules in the setting of the Deligne categories $\mathrm{Rep}(G_t)$, that was started in the previous work of the first author (arXiv:2002.01555). In this work we construct a family of Harish-Chandra bimodules that generalize simple finite dimensional bimodules in the classical case. It turns out that they have finite $K$-type, which is a non-vacuous condition for the Harish-Chandra bimodules in $\mathrm{Rep}(G_t)$. The full classification of (simple) finite $K$-type bimodules is yet unknown. This construction also yields some examples of central characters $\chi$ of the universal enveloping algebra $U(\mathfrak{g}_t)$ for which the quotient $U_\chi$ is not simple, and, thereby, it allows us to partially solve a question posed by Pavel Etingof in one of his works.