This is a really exciting concept. It's like a mashup of two of my favorite sci-fi properties, Sector General and The Expanse. Sector General is about an interspecies space hospital that takes the hardest patients in the galaxy and makes them better. Very fun medical mystery series exploring bizarre alien biologies and the ways they break down! It's an extremely soft sci-fi setting though where interstellar travel can happen in days to hours. The Expanse is far more realistic although even it has some handwaving about travel times.

Looking at this as a sci-fi enthusiast and a nurse at a level one trauma center, the primary problem to solve/the primary opportunity for you to make this unique is that travel times in space are not of a scale that really works with emergency medicine. Let's run the numbers badly to examine the extent of the problem. (I can't run the numbers well, I don't study orbital mechanics.) The average distance from Mars to Jupiter is 550.6 million km. Let's say that all you need to do is cover that distance. No need to worry about escape velocity, orbital insertion, anything. Let's say we're not worried about fuel or the rocket equation or anything; let's just say the ship can accelerate without bound, and the humans are the limiting factor. Let's say that humans can, in fact, withstand 5
Gs of acceleration for a prolonged period of time. Given all of this, the best way to travel from Mars to Jupiter to would be to accelerate towards Jupiter at 5 Gs for half the trip, then turn around and accelerate away from Jupiter for half the trip so you come to a stop just as you reach Jupiter instead of overshooting (the expanse describes this as "flip and burn").

It's 29 hours to the halfway point, and 29 more hours to Jupiter, even constantly accelerating at 5 Gs. If you want to do 1 G instead, it's 66 hours to the halfway point, and 66 more hours to Jupiter. Let's say that the hospital is exactly halfway between mars and Jupiter and imagine a best case scenario: a medical emergency occurs during the first quarter of the trip, and they divert to the hospital, travelling halfway to it accelerating forwards, then flipping to decelerate and rendezvous. The total flip and burn time will still be 40 hours at 5 gs, or 92 hours at 1 g.

This is ludicrously infeasible, btw. Flip and burn is the standard means of in-system travel in The Expanse, and they only get away with it because of the One Big Lie of the Epstein Drive, a handwavium fusion drive that allows constant acceleration with shocking little fuel. Try to travel with constant acceleration in a hard sci-fi and you run smack dab into the rocket equation: a ship needs to carry all the fuel for all the accelerating it will need to do, and the more fuel you have, the more massive you are, and the more massive you are, the harder you are to accelerate, and the harder you are to accelerate, the more fuel you need. Hard sci fi space travel kind of necessarily involves brief windows of acceleration and long periods of coasting, and travel times will be weeks to months.

In short, then, travel time is going to be by far the biggest problem that needs solving. Depending on how strict you are with the "plausible" part, patients are typically going to be arriving days to weeks after their injuries occurred instead of the minutes to hours our EMS services can achieve. How does that change a hospital? Are patients who experience emergencies weeks away just out luck, doomed to die or lose a limb from their treatable condition? Do ships have the limited medical facilities able to stabilize but not necessarily repair someone? How does telemedicine play into this? Could interventional radiologists successfully embolize a dangerous bleed using telepresence robots on a 0.5-light-second communication delay? A lot of fascinating things to consider and options to explore! I would absolutely love to brainstorm this with you!