You can read the discussion here Render names of building:part · Issue #3188 · gravitystorm/openstreetmap-carto · GitHub (and other linked discussions there).
But from what I’ve gathered
- building:part is an incorrect way to define a hospital block as main building:part purpose is to be used in 3d mapping.
- There is no currently accepted way to tag a hospital block. You may want to read through this, maybe the answer is there, I haven’t read it Tagging healthcare:speciality=* inside a hospital