As a long-time resident of Oxford-On-Rideau, I have never seen the South Branch River/Kemptville Creek having anything approaching “River” status. Most of it is regularly just a series of unconnected puddles upstream from Kemptville. I have no sentimental attachment to the name, but it should not have the word “river” as part of the name.
Perhaps a section of the water course may be called Kemptville Creek upstream from Kemptville. The Kemptville sections may be called something like “Rideau River – South Branch Inlet” or “South Branch Rideau Inlet”.
The motorboating public may be inclined to dock near Kemptville if the name “Inlet” suggests just a short excursion from their route along the Rideau system. To imply that they may travel a good distance upstream without a portage is disingenuous.