Can I branch transfer between Aviation and Medical Corps to get my flight surgeon qualification then back to Aviation to fly?

Probably possible, however:
1. If you think that is a force multiplier deal, you are mistaken: if you are branched AV in a pilot slot you cannot practice medicine in the Army, and if you are branched MC in a surgeon slot, you cannot pilot an aircraft, you can only do one of those jobs at a time. At most that is a future option multiplier.

2.AMEDD can be real stingy on letting doctors go, (to any job, no matter how critically short that may be, even 18A), and your state might not see the value in losing a surgeon just to let him go back to driving a bus, which is literally how they would see it.

And joes are lined up around the block to go to flight school, but not so much for being a doctor in uniform.

In the past, students attending the Army flight surgeon course used to learn to operate the aircraft, at least on a very basic level, but they did away with that in the mid 1990s, as I understand it. When I went, we got a little bit of simulator time, which was REALLY fun (and you'll impress your buddies I'm sure while they're all trying to land successfully at least once), but that was it. It is now a 6 week course done at Ft. Rucker, and with your pilot experience, most of it should be fairly easy for you.

A lot of what you'll cover will already be familiar to you. Army aviation history, aircraft familiarization, flight physiology...all of this you'll have been exposed to, and while most of the other docs are trying to understand concepts like dynamic rollover, you'll be fine. You'll get to topics like flight physicals, ARMS inspections, accident investigation, APLs, etc that might be new to you, but not too difficult to grasp. You don't learn medicine at the course, and you're not meant to. The real purpose is to teach you how to function as a medical advisor to the command in an aviation unit, and how to medically manage that unit's personnel.

As far as trying to be a pilot AND a doc, that might be tough. I agree with the others that both are valuable assets to the Army that take a lot of time and effort to train, but trying to balance you clinical/administrative time, with your flight time as a flight surgeon where you'll be training with your crew in that capacity, training flight medics, etc AND flying as the pilot might be tough.

Another thing to consider is your desired medical pathway. Are you interested in the Army's Residency in Aerospace Medicine? That's unquestionably more in depth training and expertise, and if you want to progress in the aviation medicine community, you'll need that. 61Ns are soon going to be reserved strictly for RAM graduates, and the rest of us who do the 6 week course with get the M3 ASI (or so I am told). If you go the RAM route, that would likely be easier to try and continue to get flight hours, even as a pilot, that if you pursue another medicine specialty. Reason being is that although you can get a flight surgeon billet (usually at the battalion level) with a number of different medical backgrounds, you'll eventually move out of that position and on to jobs that demand your specific subject matter expertise. They seem to really want RAMs, moving forward, for brigade and higher level positions.

Best of luck!

Back in the 1980s in my Army Reserve flight facility we had a flight surgeon that did our flight physicals. He would drive to the facility after his office hours and fly with us during the flight sessions. I remember that he got into the right seat and fly with the controls for an hour and then we changed seats. He ended up with his senior flight surgeon wings and retired as a Major General. Check the regulations. :-)