Keen to follow this, and I hope the OP will share progress and results.
My first Lotus was a 2009 Elise, which I was able to purchase new. Although I'd had sports cars in the past, my direct transition to the Lotus was from a "luxury" SUV, where the engine was approximately 3 km away, so to have the engine about 12 inches behind my ear. separated only by a thin sheet of glassfibre below, and a thin sheet of clear glass above, was quite a change. Being new, the issue wasn't rattles, but simply loudness and droning.
I removed every part of the inside, and placed foam insulation blocks in some areas, and Dynamat everywhere I could: floors, inside doors, especially on the rear bulkhead (engine i back, so front bulkhead not much of an issue, but as the dash was mostly out, did what I could back there too). The Lotus featured acres of very had plastic, and plain painted metal, mostly aluminium, so I also covered all of these plastic parts with leather (had this done by the Lotus Trim Shop, which no longer exists, I believe). Previous to this, it was not possible to have much conversation, or hear the radio at all, and post this, conversation and music became possible, the car was just overall quieter, could hear the stereo, and droning substantially reduced. This had been a car that, if you planned to drive more than an hour, you really needed to wear earplugs.
I don't expect this kind of result with my recently acquired '64 Plus 4, given it's age and construction type versus a modern car, even a Lotus, but I can't imagine there wouldn't be some improvement. My hesitation has been that once stuck, Dynamat - and similar products - are quite difficult to unstick. The Lotus was mostly a one-piece tub, whereas the Morgan is many (many) separate parts, which will likely need removing at some point. I've thought about starting with just a layer of their very heavy padding, which doesn't require adhesive. Don't know if anyone else has tried this.