Off the top of my head, as regards laying carpet on top of a wooden floor, I would say no, because the wooden floors would conduct sound downwards. But I’m not an expert. Did you get expert advice as suggested?
More generally – it is understandable that you have had a lot of stress and you want a resolution. The question is, do you trust the other owner, taking into account what has occurred to date? I wouldn’t, but that’s me. Even if you do negotiate with the owner, there are a lot of things up for debate, starting with where he is going to lay the carpet.
If the owner has contacted you to settle, this may be an indication that he believes his chances at the CTTT are slim. If you want to have a resolution that deals with all your concerns, I would be inclined to proceed with the CTTT, on the basis that you will likely succeed and will then have an order that the owner can’t avoid.
if you don’t want to do that, at the very least get a lawyer to draw up a legally binding agreement, because otherwise any resolution you reach with this owner will be as useful as a chocolate teapot. Asking people on a forum to do that sort of thing is not a good idea.