If it was built in the 1980s I presume it will have the original Triumph chassis rather than the later bespoke chassis.
The DVLA classify these "Kit conversions" effectively as a "rebody" of the original car as long as the original chassis has not been altered. There is an excellent thread explaining it here
http://www.madabout-kitcars.com/foru...+modifications
Also refer to the DVLA's website
https://www.gov.uk/vehicle-registrat...erted-vehicles and the information leaflet INF26
You should be able to get the cars description on the V5C changed without the need for an IVA test. The more documentation you have the better, receipts for the original kit and donor car, pictures of the donor car showing the registration number and so on.