I'm new to this Forum lark, but having read these posts thought my experience might be relevant.
My 1969 13/60 convertible had exactly the same vibration when I bought it and I spent the first two years and loadsamoney trying to find the cause. Everyone said it would be the propshaft, but I had that balanced twice - each time we did something else it seemed to improve it a bit but didn't cure it until finally someone asked if I had the strap prop. I'd never heard of that, so he explained, we looked and it did have it. I bought a second hand solid one from the Spitfire Graveyard which cured it instantly! It was a little noisy, so when one of the U/Js needed replacing I bit the bullet and got a new sliding spline shaft from Canley Classics, by which time the price had gone up considerably of course. Apparently the strap shafts can be fine when on the balancing machine but once back on the car they presumably do what they're designed for and move so, in effect they can't be balanced.
I now have that, which anyone can have for nowt if it's any use, and the solid one, which anyone can have for what it cost me (£26 I think) or I'll keep it as a spare. Hope that's helpful. Peter