I think maybe you are looking at this backwards:
We agree that logically the shuttle bay should be near the hospital, and we agree that the thing near the top-rear of the secondary hull looks like a shuttle bay, yet for some reason you think the hospital needs to be far away from there?
If we use the "ball" as quarters for the crew and the location for as many vital systems as possible, it frees up a lot of space in the secondary hull for a hospital.
The warp core you've used looks like it would fit horizontally in the "hump" between the nacelles, for example.