Unitarizable highest weight modules are Harish-Chandra modules which are at the same time simple quotients of (generalized) Verma modules. They were classified in the eighties. The title sums up the question:
Q: What are the $K$-types of unitarizable highest weight modules?
Please note that the group $K$ is also the Levi subgroup of the relevant parabolic subgroup which defines the generalized Verma modules in question. Hence the question can be rephrased as the branching of simple quotients of generalized Verma modules with respect to the Levi subgroup.
I've searched the literature and found only the Hua-Kostant-Schmid formula which gives the result for holomorphic discrete series modules with scalar minimal $K$-type. There are some papers by Toshiyuki Kobayashi on branchings of these beasts, but nothing explicit apart from the already mentioned HKS formula. I suppose there can be some results in the physics literature.