I don't believe that a 1985 CVH Morgan had a catalytic converter yet. At least my 1984 does not have one.
With the low mileage and the description above I would also assume it is fuel related. A quick an easy check is the AutoChoke on the Weber 32/34 DFT. During my restoration I bought 3 old replacement units - all over them had a broken weight on the choke lever arm. This causes strange issues ...
I had a couple of blog posts on the Weber during my restoration. There is a picture with red arrow in the gallery of
this post. Have a look.
Before you unscrew the 3 screws of the thermostat housing - use a punch or paint to mark the location. You want to install it back in the same location if the inside still looks ok.
The low mileage (and maybe the use of E10 fuel) could also mean the jets and carb need to be cleaned. Have a look
at the Webcon facebook page When you remove the jets - make sure to remember the correct location for the primary and secondary jets.
Contact Juan in the UK (from the youtube channel Air Fuel Mixologist) if you don't want to do a carb cleanup/rebuild yourself.
Even though I spent money and time on the Weber 32/34 DFT - I still consider replacing it with a 32/34 DMTL. Despite comments and recommendations here in the forum for a Weber 32/36 DGV - Webcon told me:
..." 32/36 DGV is designed to suit the Pinto and the DMTL is designed to suit the CVH, be it Transverse in the XR3 or longitudinal in the Morgan. "Hope this helps somehow
Regards
Tim