Did the Grand Army of the Republic have a specialised medical division with its own ranks? I know there's regular field medics and medical officers, there's also a 'chief medic'. Would it be plausible to assume there's a Clone Surgeon General and other such ranks?
I couldn't find anything in the canon to suggest that such a structure/role existed. The only medical structure that I could ascertain is along the lines of what you have listed, starting with clone trooper medics and various medical droids (FX- and IM-6 series) in the field, alongside Republic Mobile Surgical Units, and then clone medical officers serving aboard one of the 20 Medical Centers that the Republic had (Kaliida Shoals, Felucia and Ord Cestus were three of them). Each center could serve up to 80,000 clones at a time, and they each worked for a Sector Army which was headed up by a senior Jedi general.
In The Clone Wars tv series, Kaminoans were always shown as being in charge of dealing with the clones' medical issues, such as when Kaliida Shoals was attacked by Grievous with the Malevolence in TCW 1x3 and Nala Se was the administrator of the center, or when the clone Tup malfunctioned and Fives investigated the inhibitor chips in TCW 6x1-4. Though Kit Fisto was temporarily stationed at Ord Cestus and Nala Se worked under him during that time.
The Jedi had their own medical structure. They had the Medical Corps which was part of the Service Corps of the Jedi Order. The Medical Corps was organised by the Circle of Jedi Healers (made up of Jedi healers, headed by the Chief Healer) both of whom operated from the Halls of Healing in the Jedi Temple. Both Medical Corps-people and Jedi Healers also worked out in the field when necessary, for instance when Jedi Healers were dispatched to Kaliida Shoals to heal a virus that Cad Bane had tried to use on the clones there.
The Jedi medical structure was run by Jedi and the GAR medical centers were overseen by Jedi and served the armies run by the Jedi. The only people who really demonstrated either the aptitude or the profession to treat the clones were the Kaminoans and the clones themselves. I can't find any evidence that anyone else was involved.