I did a little bit of digging - comparing the Server tests, which run fine executed through the API, with the Client tests, which are 'not found' when trying to execute them from the API.
This is the IIS directory structure from the Server tests (top one) and the Client tests (bottom one).

The directory of the Client tests are completely missing a "Blocks" directory - and that's at least where the Server tests are run from successfully.
So the API error message of "Could not find TestSuite" for the Client tests make sense to me - they are not stored/generated in the IIS directory (as the Server tests are inside the 'Blocks') directory...
So the question I should be asking is probably "Why are the client tests not generated in the IIS directory?" - What could be missing?