What kind of noun is the 'Royal College of Surgeons'?
Solution 1:
It is a proper name - an expression which has been conventionally adopted as the name of a particular entity (CaGEL p515). These differ from proper nouns in that they have the status of noun phrases: they are not just a single noun like Thames which cannot be used in places where a noun phrase would be required like subject or object.
When in the genitive case, proper names with post-head dependents are phrasal genitives because the genitive marking is located on the last word of the post head dependent of Surgeons, and not on the head noun College (CaGEL p479).
Since the genitive is associated with the regular noun surgeons, the bare genitive is obligatory - just the apostrophe with no s following (CaGEL p481).