I've seen some great answers to the question as asked. I want to also toss the new-Trek engineers in there: Stamets, Billups, Jankom, and Pelia.

• Jankom would be awful. Terrible bedside manner and a predilection for percussive maintenance do not make for good medical outcomes.

• Billups would likely not do well either. Granted, we don't know much about him, but it seems like he's a competent but not outstanding chief engineer (let's face it, that's kinda the point of the show).

• Stamets would freak. He would repeatedly note that it's actually his husband who's the doctor. But he'd probably end up doing a decent job.

• Pelia has the distinct advantage of centuries of experience in her favor. It's entirely possible that she was a doctor at one point.

And, honorable mentions:

• Hemmer would probably shock everyone by being amazing at the job. He would probably show some unexpected advantages to being blind as a doctor.

• Jett Reno actually did this scenario, in her first episode.

Since we already know, thanks to Pulaski's ageing incident, and the Picard, Ro, Guinan Kid incident, that cures are as easy as stamping a previous pattern onto them during transport, technically any one of them would be better than an actual doctor.

Throw them into the pattern buffer, upload the last stored healthy pattern from their most recent beam-out. voila...all fixed.

Geordi, his visor can pick up on all sorts of things. He's operated on Data so often, and has often shown a capacity to apply theories across disciplines.

Scanning bone breaks and seeing them as engineering puzzles, repairing ruptures, understanding stresses, seeing heart beats and heat signatures....

...100% Geordi hands down. Plus he's good with people.

Of course in terms of real life, James Doohan (Scotty), as he's a WW2 veteran.

I think we can pretty safely put Scotty and Trip at the bottom of the pile. They've never shown any specific aptitude for medical work and I think 24th century Starfleet basic medical training and technology helps everyone else. Of the remaining three, Geordi seems like he would be very nice and try to be as helpful as possible about your injuries, but I'm ranking him 3rd overall. I don't recall O'Brien ever doing anything on screen to indicate his medical competence, but we can guess that maybe he's seen enough combat to have picked up some of the basics, so second place to Miles. Torres, with her extensive knowledge of programming and reprogramming the Doctor (enough to suggest a DNA re-sequencing of her own baby to the Doctor [Lineage, Voy]) and the fact that her year at the academy may have been a double major in engineering and medical [Extreme Risk, Voy] makes me put her in first place.

Somebody just watched the second episode of Lower Decks.

O'Brien, the hero of Setlik 3, would be fighting. Torres would be as well.

Geordi would probably be the better medic.

Reno did pretty well for an engineer when she kept the Hiawatha crew alive. She didn't really treat any of them, but she approached the situation from an engineering standpoint since the body is basically a machine, and she kept the machine running.

Of the ones listed, Scotty and O'Brien would be the most likely to just store the injured in the pattern buffer for later, proper medical treatment (like Dr. M'Benga did in SNW), so I would say they would perform the best.

Edit: Didn't see your transporter bit. I still think Scotty and/or O'Brien would improvise something along those lines and solve the problem with the pattern buffer.