Speaking with my hovercraft hat on - definitely fictitious.
They did have a dedicated liveried bus link between Dover Priory and the Hoverport, plus Calais Ville and the Hoverport, though - but this was basically just a vinyl of the HS name on the livery of whatever bus company they hired the vehicles in from!
Plus, I'm sure, some Hoverspeed branded coaches of the road vehicle type running between London and Dover Hoverport. Not sure if they ever had a matching operation on the Calais side to Paris, though.
Also, when Hoverspeed were working out of Dover/Folkestone?/Ramsgate, you'd be lucky to get a 12-car 4-CEP + MLV as a boat train, let alone something as posh as a Mk3, which were all basically dedicated to the WCML/Scotrail Glasgow-Edinburgh shuttles).