It doesn't need to be weighed, it looks like the muppet you were dealing with doesn't know his/her job.
When you do get the Registration Certificate make sure the following fields have the correct information.
F.1 should be 3,500kg (if that's what's on the VIN Plate and the CoC)
After what you've been put through
G might have the weight from the weighbridge, but it should be blank as it says (Not M1) although it is in reality irrelevant.
J should be M1
D.3 and
J.2 should be Motor Caravan
S.1 should have the number of travel (seat belted) seats, as in the CoC.
The above are the details that must be correct as if they are not it could lead to difficulties in any interactions with the law or insurance in the event of an accident.
Incorrect details at
F.1,
J, and
J.2 could also affect toll charges where the National Vehicle and Driver File (NVDF) is the source of information for determining the toll rate.