Well it seems that the DVLA on initial issue of the V5/V5c entered the vehicle type/details how they saw fit. As Graham rightly said they are in effect all 4/4's. Some being classified by their engine size, some by type, sub category or just plain 4/4 and cc rating. They have never shown chassis size or body width deviations etc. and you certainly are not the first to come across the job's worth's over this!
Use a recognised Morgan insurer, google it or if you join MSCC in the monthly mag Miscellany the relevant insurers advertise. If you want to speed things up though ,Gott and Wynn is an excellent starting point. They have a man who specifically deals with Morgan's and he really knows his stuff,, their quote will also be the most competitive that you'll probably get as well!
Morgan have been registering their 4 wheel cars since 1936 so you'd think the DVLA would have got it by now! surely they would have a data base they could refer too! If I were you I wouldn't bother trying to alter anything V5C, wise just get it insured and carry on! to poke the machine who's computers always say NO! is potentially the loss of a years motoring while you battle it out with them trying to explain the impossible!