I found the "unofficial" checker after the TFL site kept bringing up one of mine as having to pay. That particular vehicle is manufactured 1972, but registered in 1982, and is tax exempt, so I thought maybe that was the reason. This thread made me check again, and now it comes up ULEZ exempt on...
I've always used cork, but that was from years back to more recently, when new offerings have come on the scene, which I've not had experience of, and have heard some reports that they're not all they are cracked up to be.
All or any of the following: Selector striker lever, selector rod bushes, gearlever spherical seat, and gearlever, that's about it for parts, but the gearlever retainers incorrectly assembled can also cause the same problem, so far better to look at the car before ordering parts.
The rear band strut may have not been located correctly and broken the band, but the rear band will only give you a loss of REVERSE and engine braking in "1", it won't affect forward drive in any other way.
^^^ You could well be right. That part has changed since I looked into it a good while back. Same as the other changes you can now make for "safety and economy" that don't affect things in the same way they used to.
AFAIK you shouldn't have had to, the engine is of the same type as the original, so no proof needed, the extra capacity just means you lose the MOT exempt status. But, most of the monkeys at DVLA haven't got a clue about the rules anyway.