Why write "Doctor of Medicine" and not "Physician" or "Physician and Surgeon", that is what I would expect, curious. Dr of Medicine raises a question, did he complete his education and was he working as a Licentate?
Working on the basis that he was a full member of a College then you have his name and because it was a regulated profession it is reasonable to expect there to be records.
I suggest that you ignore the schools of medicine for now and look for the promulgation of fellowship. The Times is a good starting point (are they on line, I don't know).
What you should bear in mind is that the examining college was not necessarily in the country of one's medical school - you were just trying to get your fellowship any way possible.
When you know the College and date then you can focus on the records of that College (I know not what details they set out).
Even if he was a Licentate my guess is the College who granted it has records, but which College and where? Not sure if these are also promulgated or not.
Happy hunting
Canuc