Tom Regner deserves primary credit here for prompting the enquiry that produced this answer. I am posting my findings here as an answer to help others who might run into this.
I don't know if this is a fuerte bug, shortcoming or just an api caveat that wasn't clear to me... BUT...
In order for the '/_db/[DBNAME/' prefix in an endpoint (eg full endpoint '/_db/[DBNAME/_api/cursor') to be registered and used in the header of a ::arangodb::fuerte::Request, it is NOT sufficient (as of arangodb 3.5.3 and the fuerte version available at the time of this answer) to simply call:
std::unique_ptr<fuerte::Request> request;
const char *endpoint = "/_db/[DBNAME/_api/cursor";
request = fuerte::createRequest(fuerte::RestVerb::Post,endpoint);
// and adding any arguments to the request using a VPackBuilder...
// in this case the query (omitted)
To have the database name included as part of such a request, you must additionally call the following:
request->header.parseArangoPath(endpoint);
Failure to do so seems to result in an error about an 'unknown path'.
Note 1: Simply setting the database member variable, ie
request->header.database = "[DBNAME]";
does not work.
Note 2: that operations without the leading '/_db/[DBNAME]/' prefix, seem to work fine using the 'current' database. (which at least for me, seems to be stuck at '_system' since as far as I can tell, there doesn't seem to be an endpoint to change this via the HTTP REST Api.)