While I'm all for doing something different, having such a specialised ship could be very limiting in the types of stories that could be told (mainly due to the level of medical technology that exists out there and what we've seen our frontier doctors manage to do onboard non-medical specialist ships). One concept I have thought about before that could work, with a lot of drama in the mix, parallels with the current global situation:
A species hostile to the Federation (lets say the Cardassians) are stricken by a plague, which is always lethal and seems to be unstoppable. They manage to keep the lid on it for a while until one of the researchers working on finding a cure breaks radio silence and calls for help. After some negotiations between the two governments, the Cardassians reluctantly agree to permit a single medical ship to enter their space to provide assistance. Alone in hostile space, the crew discover things are much worse than the Cardies led them to believe, now they are in a race against time to discover a way to stop the plague before it's too late. However, as they work, the team uncover that the plague isn't a natural one but something engineered and unleashed on the galaxy...
Whatever direction such a story went on, a short run of a season or two would most likely work best, otherwise it could start getting a little repetitive over a longer run.
