I naturally contacted the largest Morgan dealerships in England.

The dealerships didn't have the part in stock. They contacted Morgan, who told them I had already made a request directly to the factory.

The factory replied that they didn't have the part in stock but that they would contact the Chinese supplier who had manufactured it. It's very customer-friendly of them to tell me that, but I think we shouldn't get our hopes up about finding this part.
Since this part is now 20 years old, I think the solution might come from a former enthusiast of modules made with electronic components that must have been quite basic at the time.